Crash Course Coordinate English

COORDINATE-GEOMETRY

Mathematics

RACTANGULAR CARTESIAN COORDINATES

Introduction

Co-ordinates of a point are the real variables associated in an

order to a point to describe its location in some space. Here

the space is the two dimensional plane.

The two lines XOX’ and YOY’ divide the plane in four

quadrants. XOY, YOX’, X’OY’, Y’OX are respectively called the

first, the second, the third and the fourth quadrants. We

assume the directions of OX, OY as positive while the

directions of OX’, OY’ as negative.

X

Y'

X'O

Y

Quadrant II Quadrant I

Quadrant IVQuadrant III

Cartesian co-ordinates of a point

This is the most popular co-ordinate system.

Axis of x : The line XOX’ is called axis of x.

Axis of y : The line YOY’ is called axis of y.

Co-ordinate axes : x axis and y axis together are called

axis of co-ordinates or axes of reference.

Origin : The point ‘O’ is called the origin of co-ordinates

or the origin.

Let OL = x and OM = y which are respectively called the

abscissa (or x-coordinate) and the ordinate (or y-

coordinate). The co-ordinate of P are (x, y).

Here, co-ordinates of the origin is (0, 0). The y co-

ordinates of every point on x-axis is zero.

The x co-ordinates of every point on y-axis is zero.

Oblique axes : If both the axes are not perpendicular

then they are called as oblique axes.

Polar co-ordinates

Let OX be any fixed line which is usually called the initial

line and O be a fixed point on it. If distance of any point

P from the O is ‘r’ and ��XOP , then (r, � ) are called

the polar co-ordinates of a point P.

If (x, y) are the cartesian co-ordinates of a point P,

then �cosrx � ; �sinry �

and 22 yxr �� ; ��

��

� �

x

y1tan� .

Y'

Y

X�O

�

r

P(r,�)

X

Distance formula

Y��

O

Y

P

XX

Q

R

The distance between two points ),( 11 yxP and ),( 22 yxQ

is given by

22 )()( QRPRPQ �� 212

212 )()( yyxx ��

Distance between two points in polar co-ordinates :

Let O be the pole and OX be the initial line. Let P and Q

be two given points whose polar co-ordinates are ),( 11 �r

and ),( 22 �r respectively..

r2

X

P(r1,�1)

Q(r2,�2)

M

r1

�2

(�1–�2)

�1

O

Mathematics

Then, )cos(2)( 212122

21

2 �� rrrrPQ

)cos(2 212122

21 �� rrrrPQ ,

where 1� and 2� in radians.

Properties of some geometrical figures

r2

X

P(r1,��1

Q(r2,��2

M

r1

��2

(��1-��2)

��1

(1) In a triangle ABC, if AD is the median drawn to BC, then

)(2 2222 BDADACAB ��

(2) A triangle is isosceles if any two of its medians are equal

or two sides are equal.

(3) In a right angled triangle, the mid-point of the hypotenuse

is equidistant from the vertices.

(4) Equilateral triangle : All sides are equal.

(5) Rhombus : All sides are equal and no angle is right

angle, but diagonals are at right angles and unequal.

(6) Square : All sides are equal and each angle is right angle.

The diagonals bisect each other.

(7) Parallelogram : Opposite sides are parallel and equal

and diagonals bisect each other.

(8) Rectangle : Opposite sides are equal and each angle is

right angle. Diagonals are equal.

(9) The figure obtained by joining the middle points of a

quadrilateral in order is a parallelogram.

Section formulae

If ),( yxP divides the join of ),( 11 yxA and ),( 22 yxB in the

ratio )0,(: 2121 �mmmm

(1) Internal division : If ),( yxP divides the segment AB

internally in the ratio of 21 : mm �2

1

m

m

PB

PA�

The co-ordinates of ),( yxP are

21

1221

mm

xmxmx

�

�� and

21

1221

mm

ymymy

�

��

P (x, y)

A(x1,y1)

B(x2,y2)

(2) External division : If ),( yxP divides the segment AB

externally in the ratio of 21 : mm �2

1

m

m

PB

PA�

The co-ordinates of ),( yxP are

21

1221

mm

xmxmx

�

�� and

21

1221

mm

ymymy

�

��

2 2

1 1

Some points of a triangle

(1) Centroid of a triangle : The centroid of a triangle is the

point of intersection of its medians. The centroid divides

the medians in the ratio 2 : 1 (vertex : base)

If ),( 11 yxA , ),( 22 yxB and ),( 33 yxC are the vertices of a

triangle. If G be the centroid upon one of the median

(say) AD, then AG : GD = 2 : 1

1

EF

D

2

1 1

G2

A(x1,y1)

2

C(x3,y3)(x2, y2)B

� Co-ordinate of G are ��

��

��

3,

3321321 yyyxxx

(2) Circumcentre : The circumcentre of a triangle is the

point of intersection of the perpendicular bisectors of

the sides of a triangle. It is the centre of the circle which

passes through the vertices of the triangle and so its

distance from the vertices of the triangle is the same

and this distance is known as the circum-radius of the

triangle.

(x2,y2)B

A(x1,y1)

C(x3,y3)

EF

D

O

Let vertices A, B, C of the triangle ABC be ),(),,( 2211 yxyx

and ),( 33 yx and let circumcentre be O(x, y) and then (x,

y) can be found by solving 222 )()()( OCOBOA ��

i.e., 22

22

21

21 )()()()( yyxxyyxx ��

23

23 )()( yyxx ��

If a triangle is right angle, then its circumcentre is the

mid point of hypotenuse. If angles of triangle i.e., A, B, C

and vertices of triangle ),(),,( 2211 yxByxA and ),( 33 yxC

are given, then circumcentre of the triangle ABC is

��

��

��

��

��

CBA

CyByAy

CBA

CxBxAx

2sin2sin2sin

2sin2sin2sin,

2sin2sin2sin

2sin2sin2sin 321321

Mathematics

(3) Incentre : The incentre of a triangle is the point of

intersection of internal bisector of the angles. Also it is a

centre of a circle touching all the sides of a triangle.

A(x1, y1)

I

F E

C(x3, y3)D(x2, y2)B

a

cb

Co-ordinates of incentre

��

��

��

��

��

��

cba

cybyay

cba

cxbxax 321321 ,

where a, b, c are the sides of triangle ABC.

(4) Excircle : A circle touches one side outside the triangle

and other two extended sides then circle is known as

excircle. Let ABC be a triangle then there are three

excircles with three excentres. Let 321 ,, III be the

centres of ex-circles opposite to vertices A, B and C

respectively. If vertices of triangle are ),,( 11 yxA

),( 22 yxB and ),( 33 yxC , then

3 2

1

��

��

��

��

��

��

cba

cybyay

cba

cxbxaxI 3213211 , ,

��

��

��

��

��

��

cba

cybyay

cba

cxbxaxI 321321

2 , ,

��

��

��

��

��

��

cba

cybyay

cba

cxbxaxI 3213213 ,

Angle bisector divides the opposite sides in the ratio of

remaining sides e.g.b

c

AC

AB

DC

BD�� .

Incentre divides the angle bisectors in the ratio

bacacb :)(,:)( �� and cba :)( � .

Excentre : Point of intersection of one internal angle

bisector and other two external angle bisector is called

as excentre. There are three excentres in a triangle. Co-

ordinate of each can be obtained by changing the sign

of a, b, c respectively in the formula of in-centre.

(5) Orthocentre : It is the point of intersection of

perpendiculars drawn from vertices on opposite sides

(called altitudes) of a triangle and can be obtained by

solving the equation of any two altitudes.

FD

E

O

C(x3, y3)(x2, y2)B

(x1, y1)A

Here O is the orthocentre since

BCAE� , ACBF� , ABCD� , then

ABODACOFBCOE �� and,

Solving any two we can get coordinate of O.

The orthocentre of the triangle ABC is

��

��

��

��

CBA

CyByAy

CBA

CxBxAx

tantantan

tantantan,

tantantan

tantantan 321321

If a triangle is right angled triangle, then orthocentre is

the point where right angle is formed.

Area of some geometrical figures

(1) Area of a triangle : The area of a triangle ABC with

vertices ),(),,( 2211 yxByxA and ),( 33 yxC . The area of

triangle ABC is denoted by ‘�’ and is given as

1

1

1

2

1

33

22

11

yx

yx

yx

�� )()()((2

1213132321 yyxyyxyyx ��

In equilateral triangle

(i) Having sides a, area is 2

4

3a .

(ii) Having length of perpendicular as ‘p’ area is 3

)( 2p .

(2) Collinear points : Three points ),,(),,( 2211 yxByxA

),( 33 yxC are collinear. If area of triangle is zero, then

(i) 0�� 0

1

1

1

2

1

33

22

11

�

yx

yx

yx

� 0

1

1

1

33

22

11

�

yx

yx

yx

(ii) ACBCAB �� or ABBCAC �� or BCABAC ��

(3) Area of a quadrilateral : If ),(),,(),,( 332211 yxyxyx and

),( 44 yx are vertices of a quadrilateral, then its area

)]()()()[(2

14114344323321221 yxyxyxyxyxyxyxyx ��

Mathematics

(4) Area of polygon : The area of polygon whose vertices

are )),....(,(),,(,),( ,332211 nn yxyxyxyx is

|)}(....)(){(|2

11123321221 nn yxyxyxyxyxyx ��

Or Stair method : Repeat first co-ordinates one time in

last for down arrow use positive sign and for up arrow

use negative sign.

Area of polygon = |

::

::|2

1

11

33

22

11

yx

yx

yx

yx

yx

nn

|)}....()....{(|2

11322113221 xyxyxyyxyxyx nn ��

Transformation of axes

(1) Shifting of origin without rotation of axes : Let

),( yxP � with respect to axes OXX and OY.

Let ),(' ��O with respect to axes OX and OY and let

)','( yxP � with respect to axes O’X’ and O’Y’, where OXX

and O’X’ are parallel and OY and O’Y’ are parallel.

Then �� ',' yyxx or �� yyxx ','

Thus if origin is shifted to point ),( �� without rotation of

axes, then new equation of curve can be obtained by

putting ��x in place of x and ��y in place of y..

Y

y�

x�

P(x, y)(x', y')

X�O�

O

Y�

(�,�)

X

(2) Rotation of axes without changing the origin : Let

O be the origin. Let ),( yxP � with respect to axes OX

and OY and let )','( yxP � with respect to axes OX� and

OY� where �� '' YOYOXX

then �� sin'cos' yxx ��

�� cos'sin' yxy ��

and �� sincos' yxx ��

�� cossin' yxy ��

The above relation between ),( yx and )','( yx can be

easily obtained with the help of following table

�x �y

�'x

�'y

�cos

�sin�

�sin

�cos

(3) Change of origin and rotation of axes : If origin is

changed to ),(' ��O and axes are rotated about the new

origin 'O by an angle � in the anti-clockwise sense

such that the new co-ordinates of ),( yxP become

)','( yx then the equations of transformation will be

�� sin'cos' yxx �� and �� cos'sin' yxy ��

Y'

O

�

�X'

X

O�

Y

P(x, y)

(x', y')

(4) Reflection (Image of a point) : Let ),( yx be any point,

then its image with respect to

(i) x axis � ),( yx � (ii) y-axis � ),( yx�

(iii) origin � ),( yx �� (iv) line xy � � ),( xy

Locus

The curve described by a point which moves under given

condition or conditions is called its locus.

Equation to the locus of a point : The equation to the

locus of a point is the relation, which is satisfied by the

coordinates of every point on the locus of the point.

Algorithm to find the locus of a point

Step I : Assume the coordinates of the point say (h, k)

whose locus is to be found.

Step II : Write the given condition in mathematical form

involving h , k.

Step III : Eliminate the variable (s), if any.

Step IV : Replace h by x and k by y in the result obtained

in step III. The equation so obtained is the locus of the

point which moves under some stated condition (s).

1. The new coordinates of a point (4, 5), when the origin is

shifted to the point (1,–2) are

(a) (5, 3) (b) (3, 5)

(c) (3, 7) (d) None of these

Mathematics

2. Without changing the direction of coordinate axes, origin

is transferred to ),( kh , so that the linear (one degree)

terms in the equation 76422 �� yxyx =0 are

eliminated. Then the point ),( kh is

(a) (3, 2) (b) (– 3, 2)

· (c) (2, – 3) (d) None of these

3. The equation of the locus of a point whose distance

from (a, 0) is equal to its distance from y-axis, is

(a) 22 2 aaxy �� (b) 02 22 �� aaxy

(c) 02 22 �� aaxy (d) 22 2 aaxy ��

4. Two points A and B have coordinates (1, 0) and (–1, 0)

respectively and Q is a point which satisfies the relation

�� BQAQ .1� The locus of Q is

(a) 3412 22 �� yx (b) 3412 22 �� yx

(c) 03412 22 �� yx (d) 03412 22 �� yx

5. The locus of a point P which moves in such a way that

the segment OP, where O is the origin, has slope 3 is

(a) 03 �� yx (b) 03 �� yx

(c) 03 �� yx (d) 03 �� yx

6. If the coordinates of a point be given by the equation

),cos1( �� ax �sinay � , then the locus of the point

will be

(a) A straight line (b) A circle

(c) A parabola (d) An ellipse

7. If P = (1,0), Q =(–1,0) and R =(2,0) are three given points,

then the locus of a point S satisfying the relation222 2SPSRSQ �� is

(a) A straight line parallel to x-axis

(b) A circle through origin

(c) A circle with centre at the origin

(d) A straight line parallel to y-axis

8. The coordinates of the points O, A and B are (0,0), (0,4)

and (6,0) respectively. If a points P moves such that the

area of POA� is always twice the area of POB� , then

the equation to both parts of the locus of P is

(a) 0)3)(3( �� yxyx (b) 0))(3( �� yxyx

(c) 0)3)(3( �� yxyx (d) None of these

9. A point moves in such a way that the sum of square of its

distance from the points )0,2(A and )0,2(�B is always

equal to the square of the distance between A and B.

The locus of the point is

(a) 0222 �� yx (b) 0222 �� yx

(c) 0422 �� yx (d) 0422 �� yx

10. A point P moves so that its distance from the point )0,(a is

always equal to its distance from the line 0�� ax . The

locus of the point is

(a) axy 42 � (b) ayx 42 �

(c) 042 �� axy (d) 042 �� ayx

11. The equation to the locus of a point which moves so that

its distance from x-axis is always one half its distance

from the origin, is

(a) 03 22 �� yx (b) 03 22 �� yx

(c) 03 22 �� yx (d) 03 22 �� yx

12. A point moves so that its distance from the point (–1,0)

is always three times its distance from the point (0,2).

The locus of the point is

(a) A line (b) A circle

(c) A parabola (d) An ellipse

13. The locus of a point which moves so that its distance

from x-axis is double of its distance from y-axis is

(a) yx 2� (b) xy 2�

(c) 15 �� yx (d) 32 �� xy

14. O is the origin and A is the point (3,4). If a point P moves

so that the line segment OP is always parallel to the line

segment OA, then the equation to the locus of P is

(a) 034 �� yx (b) 034 �� yx

(c) 043 �� yx (d) 043 �� yx

15. The locus of a point which moves so that it is always

equidistant from the point A(a, 0) and B (– a, 0) is

(a) A circle

(b) Perpendicular bisector of the line segment AB

(c) A line parallel to x-axis

(d) None of these

Mathematics

16. The coordinates of the points A and B are (a, 0) and

)0,( a� respectively. I f a point P moves so thatt

222 2kPBPA �� , when k is constant, then the equation

to the locus of the point P , is

(a) 02 2 �� kax (b) 02 2 �� kax

(c) 02 2 ��kay (d) 02 2 �� kay

17. If the coordinates of a point be given by the equations

�� tan,sec aybx �� , then its locus is

(a) A straight line (b) A circle

(c) An ellipse (d) A hyperbola

18. The coordinates of the point A and B are )0,(ak and

)1(,0, ��

��

k

k

a. If a point P moves so that ,kPBPA �

then the equation to the locus of P is

(a) 0)( 2222 �� ayxk (b) 02222 �� akyx

(c) 0222 �� ayx (d) 0222 �� ayx

19. The locus of a point which moves in such a way that its

distance from (0,0) is three times its distance from the x-

axis, as given by

(a) 08 22 �� yx (b) 08 22 �� yx

(c) 04 22 �� yx (d) 04 22 �� yx

20. The equation of the locus of all points equidistant from

the point (4,2) and the x-axis, is

(a) 020482 �� yxx (b) 020482 �� yxx

(c) 020842 �� xyy (d) None of these

21. The locus of the mid-point of the distance between the

axes of the variable line ,sincos pyx �� where p is

constant, is

(a) 222 4pyx �� (b) 222

411

pyx��

(c) 2

22 4

pyx �� (d) 222

211

pyx��

22. The locus of a point whose distance from the point

),( fg �� is always ‘‘a’, will be, (where 222 afgk �� )

(a) 02222 �� kfygxyx

(b) 02222 �� kfygxyx

(c) 022222 �� kfygxxyyx

(d) None of these

23. The locus of the moving point P, such that 2PA = 3PB

where A is (0,0) and B is (4,–3), is

(a) 0225547255 22 �� yxyx

(b) 0225547255 22 �� yxyx

(c) 0225547255 22 �� yxyx

(d) 0225547255 22 �� yxyx

24. A point moves such that the sum of its distances from

two fixed points (ae,0) and (–ae,0) is always 2a. Then

equation of its locus is

(a) 1)1( 22

2

2

2

��

�ea

y

a

x(b) 1

)1( 22

2

2

2

��

�ea

y

a

x

(c) 1)1( 2

2

22

2

�� a

y

ea

x(d) None of these

25. A point moves in such a way that its distance from (1,–2)

is always the twice from (–3,5), the locus of the point is

(a) 013144263 22 �� yxyx

(b) 013144263 22 �� yxyx

(c) 01314426)(3 22 �� yxyx

(d) None of these

26. A point moves in such a way that its distance from origin

is always 4. Then the locus of the point is

(a) 422 �� yx (b) 1622 �� yx

(c) 222 �� yx (d) None of these

27. If )0,( aA � and )0,(aB are two fixed points, then the locus

of the point on which the line AB subtends the right

angle, is

(a) 222 2ayx �� (b) 222 ayx ��

(c) 0222 �� ayx (d) 222 ayx ��

28. If A and B are two fixed points and P is a variable point

such that 4�� PBPA , then the locus of P is a/an

Mathematics

(a) Parabola (b) Ellipse

(c) Hyperbola (d) None of these

29. If A and B are two points in a plane, so that PBPA � =

constant, then the locus of P is

(a) Hyperbola (b) Circle

(c) Parabola (d) Ellipse

30. If A and B are two fixed points in a plane and P is another

variable point such that �� 22 PBPA constant, then the

locus of the point P is

(a) Hyperbola (b) Circle


1.(c) 2.(c) 3.(b) 4.(b) 5.(d) 6.(b) 7.(d) 8.(a) 9.(d) 10.(a)

11.(b) 12.(b) 13.(b) 14.(a) 15.(b) 16.(b) 17.(d) 18.(d) 19.(a) 20.(b)

21.(b) 22.(a) 23.(d) 24.(a) 25.(c) 26.(b) 27.(d) 28.(b) 29.(a) 30.(b)

Mathematics

STRAIGHT LINE

Definition

The straight line is a curve such that every point on the line

segment joining any two points on it lies on it. The simplest

locus of a point in a plane is a straight line. A line is determined

uniquely by any one of the following:

(1) Two different points (because we know the axiom that

one and only one straight line passes through two given

points).

(2) A point and a given direction.

Slope (Gradient) of a line

The trigonometrical tangent of the angle that a line

makes with the positive direction of the x-axis in

anticlockwise sense is called the slope or gradient of the

line. The slope of a line is generally denoted by m. Thus,

m = �tan .

B

Y

AX� X

O

�

Y�

B

Y

AX� X

O

�

Y�

(1) Slope of line parallel to x – axis is 00tan �� om .

(2) Slope of line parallel to y – axis is �� om 90tan .

(3) Slope of the line equally inclined with the axes is 1 or – 1.

(4) Slope of the line through the points ),( 11 yxA and

),( 22 yxB is 12

12

xx

yy

�

� taken in the same order..

(5) Slope of the line 0,0 �� bcbyax is b

a� .

(6) Slope of two parallel lines are equal.

(7) If 1m and 2m be the slopes of two perpendicular lines,

then 1. 21 ��mm .

(8) m can be defined as �tan for �� 0 and 2

�� .

Equations of straight line in different forms

(1) Slope form : Equation of a line through the origin and

having slope m is y = mx.

(2) One point form or Point slope form : Equation of a

line through the point ),( 11 yx and having slope m is

)( 11 xxmyy �� .

(3) Slope intercept form : Equation of a line (non-vertical)

with slope m and cutting off an intercept c on the y-axis

is cmxy �� .

AX '

� c

O

Y '

X

B

Y

The equation of a line with slope m and the x-intercept

d is )( dxmy �� .

(4) Intercept form : If a straight line cuts x-axis at A and the

y-axis at B then OA and OB are known as the intercepts

of the line on x-axis and y-axis respectively.

Then, equation of a straight line cutting off intercepts a

and b on x–axis and y–axis respectively is 1��b

y

a

x.

If given line is parallel to X axis, then X-intercept is

undefined.

If given line is parallel to Y axis, then Y-intercept is

undefined.

(5) Two point form: Equation of the line through the points

A ),( 11 yx and ),( 22 yxB is, )()( 112

121 xx

xx

yyyy �

�

�� .

(x2, y2)

(x1,y1)

OX

LY

A

B

Mathematics

In the determinant form it is gives as

1

1

1

22

11

yx

yx

yx

= 0

is the equation of line.

(6) Normal or perpendicular form : The equation of the

straight line upon which the length of the perpendicular

from the origin is p and this perpendicular makes an

angle � with x-axis is pyx �� sincos .

A

Y'

�X ' X

Y

p

O

BP

(7) Symmetrical or parametric or distance form of the

line : Equation of a line passing through ),( 11 yx and

making an angle � with the positive direction of x-axis is

ryyxx

��

��

�� sincos11

, where r is the distance between

the point P (x, y) and ),( 11 yxA .

1 1

The co-ordinates of any point on this line may be taken

as )sin,cos( 11 �� ryrx �� , known as parametric co-

ordinates. ‘r’ is called the parameter.

Equation of parallel and perpendicular lines to a given

line

(1) Equation of a line which is parallel to 0�� cbyax is

0�� byax .

(2) Equation of a l ine which is perpendicular to

0�� cbyax is 0�� aybx .

The value of � in both cases is obtained with the help of

additional information given in the problem.

(3) If the equation of line be cba �� cossin , then line

(i) Parallel to it, dba �� cossin

(ii) Perpendicular to it, dba ��

��

��

�

��

� �

��

�2

cos2

sin .

General equation of a straight line and its

transformation in standard forms

General form of equation of a line is 0�� cbyax , its

(1) Slope intercept form:b

cx

b

ay �� , slope

b

am ��

and intercept on y-axis is, b

cC �� .

(2) Intercept form : 1//

��

�� bc

y

ac

x, x intercept is

= ��

��

�a

c and y intercept is = �

�

��

�b

c.

(3) Normal form : To change the general form of a line into

normal form, first take c to right hand side and make it

positive, then divide the whole equation by 22 ba �

like

,222222 ba

c

ba

by

ba

ax

��

��

��

where 22cos

ba

a

�� ,

22sin

ba

b

�� , 22 ba

cp

��

Point of intersection of two lines

Point of intersection of two lines 111 cybxa �� =0 and

0222 �� cybxa is given by

��

��

�

�

�

��

1221

1221

1221

1221 ,),(baba

acac

baba

cbcbyx

��

�

�

��

�

21

21

21

21

21

21

21

21

,

bb

aa

aa

cc

bb

aa

cc

bb

General equation of lines through the intersection of

two given lines

If equation of two lines 0111 �� cybxaP and

0222 �� cybxaQ , then the equation of the lines

passing through the point of intersection of these lines

is 0�� QP � or �� 111 cybxa 0)( 222 �� cybxa� .

Value of � is obtained with the help of the additional

information given in the problem.

Angle between two non-parallel lines

If � be the angle between the lines 11 cxmy �� and

22 cxmy �� and intersecting at A. Then,

21

211

1tan

mm

mm

�

�� . If � is angle between two lines,

Mathematics

then �� is also the angle between them.

(1) Angle between two straight lines when their

equations are given : The angle � between the lines

0111 �� cybxa and 0222 �� cybxa is given by,y,

2121

2112tanbbaa

baba

�

�� .

(2) Conditions for two lines to be coincident, parallel,

perpendicular and intersecting : Two lines

0111 �� cybxa and 0222 �� cybxa are,

(a) Coincident, if 2

1

2

1

2

1

c

c

b

b

a

a��

(b) Parallel, if 2

1

2

1

2

1

c

c

b

b

a

a��

(c) Intersecting, if 2

1

2

1

b

b

a

a�

(d) Perpendicular, if 02121 �� bbaa

Equation of straight line through a given point making

a given angle with a given line

The equation of the straight lines which pass through a

given point ),( 11 yx and make a given angle � with given

straight line cmxy �� are,

)(tan1

tan11 xx

m

myy �

��

��

�.

Equations of the bisectors of the angles between two

straight lines

The equation of the bisectors of the angles between the

lines 0111 �� cybxa and 0222 �� cybxa are given

by,

22

22

222

21

21

111

ba

cybxa

ba

cybxa

�

��

�

�� .....(i)

Algorithm to find the bisector of the angle

containing the origin : Let the equations of the two

lines 0111 �� cybxa and 0222 �� cybxa . To find

the bisector of the angle containing the origin, we

proceed as follows:

Step I : See whether the constant terms 1c and 2c in

the equations of two lines positive or not. If not, then

multiply both the sides of the equation by –1 to make

the constant term positive.

Step II : Now obtain the bisector corresponding to the

positive sign i.e., 22

22

222

21

21

111

ba

cybxa

ba

cybxa

�

��

�

��.

This is the required bisector of the angle containing the

origin.

The bisector of the angle containing the origin means

the bisector of the angle between the lines which

contains the origin within it.

(1) To find the acute and obtuse angle bisectors : Let

� be the angle between one of the lines and one of the

bisectors given by (i). Find �tan . If 1|tan| �� , then this

bisector is the bisector of acute angle and the other one

is the bisector of the obtuse angle.

If |tan| � > 1, then this bisector is the bisector of obtuse

angle and other one is the bisector of the acute angle.

(2) Method to find acute angle bisector and obtuse

angle bisector

(i) Make the constant term positive, if not.

(ii) Now determine the sign of the expression

2121 bbaa � .

(iii) If 02121 �� bbaa , then the bisector corresponding

to “+” sign gives the obtuse angle bisector and the

bisector corresponding to “–” sign is the bisector of acute

angle between the lines.

(iv) If 02121 �� bbaa , then the bisector corresponding

to “+” and “–” sign given the acute and obtuse angle

bisectors respectively.

Bisectors are perpendicular to each other.

If 02121 �� bbaa , then the origin lies in obtuse angle

and if 02121 �� bbaa , then the origin lies in acute angle.

Obtuse bisector

L2

L1

Acute bisector

Length of perpendicular

(1) Distance of a point from a line : The length p of the

perpendicular from the point ),( 11 yx to the line

0�� cbyax is given by 22

11 ||

ba

cbyaxp

�

�� .

Length of perpendicular from origin to the line

0�� cbyax is 22 ba

c

�.

Length of perpendicular from the point ),( 11 yx to the

line pyx �� sincos is |sincos| 11 pyx �� .

Mathematics

(2) Distance between two parallel lines : Let the two

parallel lines be 01 �� cbyax and First Method: The

distance between the lines is )(

||

22

21

ba

ccd

�

�� .

Second Method: The distance between the lines is

)( 22 bad

��

�,

ax + by + c1 = 0

ax + by + c2 = 0

O (0, 0)

where (i) || 21 cc �� , if they be on the same side of

origin.

(ii) |||| 21 cc �� , if the origin O lies between them.

Third method : Find the coordinates of any point on

one of the given line, preferably putting 0�x or 0�y .

Then the perpendicular distance of this point from the

other line is the required distance between the lines.

Distance between two parallel lines

01 �� cbyax , 02 �� ckbykax is 22

21

ba

k

cc

�

�.

Distance between two non parallel lines is always zero.

Position of a point with respect to a line

Let the given line be 0�� cbyax and observing point

is ),( 11 yx , then

(i) If the same sign is found by putting in equation of line

11, yyxx �� and 0�x , 0�y then the point ),( 11 yx is

situated on the same side of origin.

(ii) If the opposite sign is found by putting in equation of

line ,1xx � 1yy � and 0�x , 0�y then the point

),( 11 yx is situated opposite side to origin.

Position of two points with respect to a line

Two points ),( 11 yx and ),( 22 yx are on the same side or

on the opposite side of the straight line 0�� cbyax

according as the values of cbyax �� 11 and

cbyax �� 22 are of the same sign or opposite sign.

Concurrent lines

Three or more lines are said to be concurrent lines if

they meet at a point.

First method : Find the point of intersection of any two

lines by solving them simultaneously. If the point satisfies

the third equation also, then the given lines are

concurrent.

Second method : The three lines 0111 �� cybxa ,

0222 �� cybxa and 0333 �� cybxa are concurrent

if, 0

333

222

111

�

cba

cba

cba

.

Third method : The condition for the lines 0�P , 0�Q

and 0�R to be concurrent is that three constants a, b,

c (not all zero at the same time) can be obtained such

that 0�� cRbQaP .

Reflection on the surface

Surface

Tangent

P

� � � �

RNI

Here, IP = Incident Ray

PN = Normal to the surface

PR = Reflected Ray

Then, NPRIPN ��

Angle of incidence = Angle of reflection

Image of a point in different cases

(1) The image of a point with respect to the line mirror

The image of ),( 11 yxA with respect to the line mirror

0�� cbyax be B (h, k) is given by,,

221111 )(2

ba

cbyax

b

yk

a

xh

�

��

��

�

A (x1, y1)

B(h,k)

ax+by+c = 0

(2) The image of a point with respect to x-axis : Let

),( yxP be any point and P� ),( yx �� its image after

reflection in the x-axis, then x � = x

Mathematics

y� = – y, (� O� is the mid point of P and P� )

(3) The image of a point with respect to y-axis : Let

),( yxP be any point and ),( yxP �� its image after

reflection in the y-axis, then xx ��

yy �� , (� O� is the mid point of P and P� )

P(x, y)

O�

X�

P� (x�,y�)

XO

Y�

Y

(4) The image of a point with respect to the origin : Let

),( yxP be any point and ),( yxP �� be its image after

reflection through the origin, then xx ��

yy �� ,(� O is the mid point of P, P� ).

P(x, y)

NX�

P�(x�, y�)

X

Y�

Y

O M

(5) The image of a point with respect to the line y = x :

Let ),( yxP be any point and ),( yxP �� be its image after

reflection in the line xy � , then yx ��

xy �� , (� O� is the mid point of P and P� ).

(6) The image of a point with respect to the line

y = x tan �� : Let ),( yxP be any point and ),( yxP �� be its

image after reflection in the line �tanxy � , then

�� 2sin2cos yxx ��

�� 2cos2sin yxy �� , (� O� is the mid point of P and

P� )

P(x, y)

X� X

Y�

Y

O

O�

P�(x�, y�)y=x tan ��

1. If the extremities of the base of an isosceles triangle are

the points )0,2( a and ),0( a and the equation of one of

the sides is ax 2� , then the area of the triangle is

(a) sqa 25 . units (b) .2

5 2sqa units

(c) .2

25 2

sqa

units (d) None of these

2. The equation to the sides of a triangle are 03 �� yx ,

534 �� yx and 03 �� yx . The line 043 �� yx passes

through

(a) The incentre (b) The centroid

(c) The circumcentre

(d) The orthocentre of the triangle

3. Area of the parallelogram formed by the lines

0111 �� cybxa , 0111 �� dybxa and

0222 �� cybxa , 0222 �� dybxa is

(a) 2/122

22

21

21

2211

)])([(

))((

baba

cdcd

��

��

(b)2121

2211 ))((

bbaa

cdcd

�

��

(c)2121

2211 ))((

bbaa

cdcd

�

��(d)

1221

2211 ))((

baba

cdcd

�

��

4. Area of the parallelogram whose sides are

pyx �� sincos ,sincos qyx ��

ryx �� sincos and syx �� sincos is

(a) )(cosec))(( �� srqp

(b) )(cosec ))(( �� srqp

(c) )(cosec ))(( �� srqp

(d) None of these

Mathematics

5. The area of the triangle bounded by the straight line

)0,,(,0 �� cbacbyax and the coordinate axes is

(a)||2

1 2

bc

a(b)

||2

1 2

ab

c

(c)||2

1 2

ac

b(d) 0

6. The triangle formed by the lines ,04 �� yx

,43 �� yx 43 �� yx is

(a) Isosceles (b) Equilateral

(c) Right–angled (d) None of these

7. Two lines are drawn through (3, 4), each of which makes

angle of 45o with the line 2�� yx , then area of the

triangle formed by these lines is

(a) 9 (b) 9/2

(c) 2 (d) 2/9

8. The area of the triangle formed by the line

�� 2sincossin �� yx and the coordinates axes is

(a) �2sin (b) �2cos

(c) �2sin2 (d) �2cos2

9. The area of a parallelogram formed by the lines

0�� cbyax , is

(a)ab

c 2

(b)ab

c 22

(c)ab

c

2

2

(d) None of these

10. The triangle formed by 09 22 �� yx and 4�x is

(a) Isosceles (b) Equilateral

(c) Right angled (d) None of these

11. A point moves so that square of its distance from the

point (3, – 2) is numerically equal to its distance from

the line 13125 �� yx . The equation of the locus of the

point is

(a) 018264831313 22 �� yxyx

(b) 026161122 �� yxyx

(c) 0161122 �� yxyx

(d) None of these

12. Locus of the points which are at equal distance from

01143 �� yx and 02512 �� yx and which is near

the origin is

(a) 01537721 �� yx (b) 01337799 �� yx


13. A point moves such that its distance from the point

)0,4( is half that of its distance from the line 16�x . The

locus of this point is

(a) 19243 22 �� yx (b) 19234 22 �� yx


14. The locus of a point so that sum of its distance from two

given perpendicular lines is equal to 2 unit in first

quadrant, is

(a) 02 �� yx (b) 2�� yx

(c) 2�� yx (d) None of these

15. If the sum of the distances of a point from two

perpendicular lines in a plane is 1, then its locus is

(a) Square (b) Circle

(c) Straight line (d) Two intersecting lines

16. If a variable line drawn through the point of intersection

of straight lines 1��yx

and 1��yx

meets the

coordinate axes in A and B, then the locus of the mid

point of AB is

(a) )()( �� xyyx (b) )(2)( �� xyyx

(c) xyyx �� 2))(( �� (d) None of these

17. The point moves such that the area of the triangle formed

by it with the points (1, 5) and (3, –7) is 21sq. unit. The

locus of the point is

(a) 0326 �� yx (b) 0326 �� yx

(c) 0326 �� yx (d) 0326 �� yx

18. A straight line through the point (1, 1) meets the x-axis

at ‘A’ and the y-axis at ‘B’. The locus of the mid-point of AB

is

(a) 02 �� yxxy (b) 02 �� xyyx

(c) 02 �� yx (d) 02 �� yx

Mathematics

19. If A is (2, 5), B is (4, –11) and C lies on 0479 �� yx ,

then the locus of the centroid of the ABC� is a straight

line parallel to the straight line is

(a) 0497 �� yx (b) 0479 �� yx

(c) 0479 �� yx (d) 0497 �� y

20. The number of integral values of m, for which the x-co-

ordinate of the point of intersection of the lines

943 �� yx and 1��mxy is also an integer is

(a) 2 (b) 0

(c) 4 (d) 1

21. A ray of light coming from the point (1, 2) is reflected at

a point A on the x–axis and then passes through the

point (5, 3). The coordinates of the point A are

(a) 0,5/13 (b) 0,13/5

(c) (– 7, 0) (d) None of these

22. If the co-ordinates of the middle point of the portion of

a line intercepted between coordinate axes (3,2), then

the equation of the line will be

(a) 1232 �� yx (b) 1223 �� yx

(c) 634 �� yx (d) 1025 �� yx

23. A line through )4,5( ��A meets the lines ,023 �� yx

042 �� yx and 05 �� yx at B, C and D respectively..

If ,61015

222

��

��

��

�

��

��

�

��

ADACAB

then the equation of the

line is

(a) 02232 �� yx (b) 0745 �� yx


24. The equation of perpendicular bisectors of the sides AB

and AC of a triangle ABC are 05 �� yx and 02 �� yx

respectively. If the point A is )2,1( � , then the equation

of line BC is

(a) 0401423 �� yx (b) 0402314 �� yx

(c) 0401423 �� yx (d) 0402314 �� yx

25. The medians AD and BE of a triangle with vertices

)0,0(),,0( BbA and )0,(aC are perpendicular to each

other, if

(a) ba 2� (b) ba 2��

(c) Both (a) and (b) (d) None of these

26. Let PS be the median of the triangle with vertices

)1,6(),2,2( �QP and )3,7(R . The equation of the line

passing through (1, – 1) and parallel to PS is

(a) 0792 �� yx (b) 01192 �� yx

(c) 01192 �� yx (d) 0792 �� yx

27. The equation of straight line passing through )0,( a� and

making the triangle with axes of area ‘T’ is

(a) 022 2 �� aTyaTx (b) 022 2 �� aTyaTx

(c) 022 2 �� aTyaTx (d) None of these

28. The equations of two equal sides of an isosceles triangle

are 037 �� yx and 03 �� yx and the third side

passes through the point (1, – 10). The equation of the

third side is

(a) 0313 �� yx but not 073 �� yx

(b) 073 �� yx but not 0313 �� yx

(c) 073 �� yx or 0313 �� yx

(d) Neither 73 �� yx nor 0313 �� yx

29. The graph of the function )1(cos)2cos(cos 2 �� xxx is

(a) A straight line passing through )1sin,0( 2� with

slope 2

(b) A straight line passing through (0, 0)

(c) A parabola with vertex )1sin,1( 2�

(d) A straight l ine passing through the point

��

��

� 1sin,

22�

and parallel to the x–axis

30. If the equation of base of an equilateral triangle is

12 �� yx and the vertex is (–1, 2), then the length of

the side of the triangle is

(a)3

20(b)

15

2

(c)15

8(d)

2

15

1.(b) 2.(d) 3.(d) 4.(a) 5.(b) 6.(a) 7.(b) 8.(a) 9.(b) 10.(a)

11.(a) 12.(b) 13.(a) 14.(b) 15.(a) 16.(b) 17.(a) 18.(b) 19.(c) 20.(a)

21.(a) 22.(a) 23.(a) 24.(d) 25.(c) 26.(d) 27.(b) 28.(c) 29.(d) 30.(a)

Mathematics

PAIR OF STRAIGHT LINE

Equation of pair of straight lines

(1) Equation of a pair of straight lines passing through

origin : The equation 02 22 �� byhxyax represents

a pair of straight line passing through the origin where a,

h, b are constants.

Let the lines represented by 02 22 �� byhxyax be

01 �� xmy , 02 �� xmy . Then, b

hmm

221 �� and

b

amm �21

Then, two straight lines represented by

02 22 �� byhxyax are abhyhyax �� 2 = 0 and

02 �� abhyhyax .

Hence, (a) The lines are real and distinct, if 02 �� abh

(b) The lines are real and coincident, if 02 �� abh

(c) The lines are imaginary, if 02 �� abh

(2) General equation of a pair of straight lines : An

equation of the form,

0222 22 �� cfygxbyhxyax

where a, b, c, f, g, h are constants, is said to be a general

equation of second degree in x and y.

The necessary and sufficient condition for

0222 22 �� cfygxbyhxyax to represents a

pair of straight lines is that

02 222 �� chbgaffghabc or 0�

cfg

fbh

gha

.

Point of intersection of lines represented by ax2 +

2hxy + by2 + 2gx + 2fy +c = 0

Let 0222 22 �� cfygxbyhxyax�

0222 ��

ghyaxx

� (Keeping y as constant)

and 0222 ��

�fbyhx

y

� (Keeping x as constant)

For point of intersection 0��

�

x

� and 0�

�

�

y

�

We obtain, 0�� ghyax and 0�� fbyhx

On solving these equations, we get

2

1

habafgh

y

bgfh

x

��

��

� i.e., ��

��

�

�

�

��

abh

ghaf

abh

fhbgyx

22,),( .

(3) Separate equations from joint equation: The general

equation of second degree be

0222 22 �� cfygxbyhxyax

To find the lines represented by this equation we proceed

as follows :

Step I : Factorize the homogeneous part

22 2 byhxyax �� into two linear factors. Let the linear

factors be ybxa '' � and ybxa "" � .

Step II : Add constants 'c and "c in the factors obtained

in step I to obtain ''' cybxa �� and """ cybxa �� . Let

the lines be 0''' �� cybxa and 0""" �� cybxa .

Step III : Obtain the joint equation of the lines in step II

and compare the coefficients of x, y and constant terms

to obtain equations in c’ and c” .

Step IV : Solve the equations in c’ and c” to obtain the

values of c’ and c”.

Step V : Substitute the values of c’ and c” in lines in step

II to obtain the required lines.

Angle between the pair of lines

The angle between the lines represented by

0222 22 �� cfygxbyhxyax or

02 22 �� byhxyax

is given by ba

abh

ba

abh

�

��

�

��

21

2 2tan

2tan ��

From the above formula it is clear, that

(i) The lines represented by

0222 22 �� cfygxbyhxyax are parallel iff abh �2

and 22 bgaf � or f

g

b

h

h

a�� .

Mathematics

(ii) The lines represented by 22 2 byhxyax ��

022 �� cfygx are perpendicular iff 0�� ba

i.e., Coefficient of �2x Coefficient of 02 �y .

(iii) The lines are coincident, if acg �2 .

Bisectors of the angles between the lines

(1) The joint equation of the bisectors of the angles between

the lines represented by the equation

02 22 �� byhxyax is

h

xy

ba

yx�

�

� 22

� 0)( 22 �� hyxybahx

Here, coefficient of �2x coefficient of 02 �y . Hence,

the bisectors of the angles between the lines are

perpendicular to each other. The bisector lines will pass

through origin also.

(i) If ba � , the bisectors are 022 �� yx .

i.e., 0,0 �� yxyx

(ii) If 0�h , the bisectors are 0�xy i.e., 0,0 �� yx .

(2) The equation of the bisectors of the angles between the

lines represented by 22 2 byhxyax �� + 022 �� cfygx

are given by h

yx

ba

yx ))(()()( 2 ��

�

��

, where �,

� is the point of intersection of the lines represented by

the given equation.

Equation of the lines joining the origin to the points

of intersection of a given line and a given curve

lx+my+n=0

Y

Y�

OX' X

B

A

The equation of the lines which joins origin to the point

of intersection of the line 0�� nmylx and curve

0222 22 �� cfygxbyhxyax , can be obtained by

making the curve homogeneous with the help of line

0�� nmylx , which is

0)(22

222 ��

�

��

�

��

�

��

�

��

n

mylxc

n

mylxfygxbyhxyax

Removal of first degree terms

Let point of intersection of l ines represented by

0222 22 �� cfygxbyhxyax .....(i) is ),( �� .

Here ��

��

�

�

�

��

abh

ghaf

abh

fhbg22

,),( ��

For removal of first degree terms, shift the origin to ),( ��

i.e., replacing x by )( ��X and y be )( ��Y in (i).

Alternative Method : Direct equation after removal of

first degree terms is

0)(2 22 �� cfgbYhXYaX �� ,

where abh

fhbg

�

��

2� and

abh

ghaf

�

��

2� .

Removal of the term xy from f(X,Y) = ax2+2hxy+by2

without changing the origin

Clearly, 0�h . Rotating the axes through an angle � ,

we have, �� sincos YXx �� and �� cossin YXy ��

22 2),( byhxyaxyxf ��

After rotation, new equation is

222 )sinsincos2cos(),( XbhaYXF ��

XYhab )sin(cossincos){(2 22 ��

222 )cossincos2sin( Ybha ��

Now coefficient of XY = 0. Then we get cot h

ba

22

�� .

! Usually, we use the formula, ba

h

��

22tan � for finding

the angle of rotation �. However, if ba � , we use

h

ba

22cot

�� as in this case �2tan is not defined.

Distance between the pair of parallel straight lines

If 0222 22 �� cfygxbyhxyax represents a pair

of parallel straight lines, then the distance between them

is given by )(

22

baa

acg

�

�or

)(2

2

bab

bcf

�

�.

1. The equation 4)2()2( 2222 �� yxyx

represents a

(a) Circle (b) Pair of straight lines


2. If the equation 012222 �� fygxyx represents a

pair of lines, then

(a) 122 �� fg (b) 122 �� gf

(c) 122 �� fg (d)2

122 �� gf

Mathematics

3. If the pair of straight lines 01 �� yxxy and the line

032 �� yax are concurrent, then a =

(a) – 1 (b) 0

(c) 3 (d) 1

4. The area of the triangle formed by the line

0994 22 �� yxyx and 2�x is

(a) 2 (b) 3

(c) 10/3 (d) 20/3

5. If the equations of opposite sides of a parallelogram are

0672 �� xx and 040142 �� yy , then the

equation of its one diagonal is

(a) 01456 �� yx (b) 01456 �� yx

(c) 01465 �� yx (d) 01465 �� yx

6. The image of the pair of lines represented by

02 22 �� byhxyax by the line mirror 0�y is

(a) 02 22 �� byhxyax (b) 02 22 �� ayhxybx

(c) 02 22 �� ayhxybx (d) 02 22 �� byhxyax

7. Let PQR be a right angled isosceles triangle, right angled

at )1,2(P . If the equation of the line QR is

,32 �� yx then the equation representing the pair of

lines PQ and PR is

(a) 0251020833 22 �� yxxyyx

(b) 0251020833 22 �� yxxyyx

(c) 0201510833 22 �� yxxyyx

(d) 0201510833 22 �� yxxyyx

8. If the portion of the line 1��mylx falling inside the

circle 222 ayx �� subtends an angle of o45 at the

origin, then

(a) )(]1)([4 222222 mlamla ��

(b) 2)(]1)([4 222222 �� mlamla

(c) 2222222 ]2)([]1)([4 �� mlamla

(d) None of these

9. The angle between lines joining the origin to the points

of intersection of the line 23 �� yx and the curve

422 �� xy is

(a) 6/� (b) 4/�

(c) 3/� (d) 2/�

10. Mixed term xy is to be removed from the general

equation 022222 �� cgxfyhxybyax . One

should rotate the axes through an angle � given by

�2tan equal to

(a)h

ba

2

�(b)

ba

h

�2

(c)h

ba

2

�(d)

)(

2

ba

h

�

11. If the equation 0)3(3 2323 �� xyxmyxy represents

the three lines passing through origin, then

(a) Lines are equally inclined to each other

(b) Two lines makes equal angle with x-axis

(c) All three lines makes equal angle with x-axis

(d) None of these

12. Locus of the points equidistant from the lines

represented by 0sinsincos 22222 �� yxyx is

(a) 0sec2 222 �� xyyx

(b) 0cosec2 222 �� xyyx

(c) 0sec2 222 �� xyyx

(d) 0cosec2 222 �� xyyx

13. If pair of straight l ines 02 22 �� ymxyx and

02 22 �� ynxyx be such that each pair bisects the

angle between the other pair, then mn =

(a) 1 (b) – 1

(c) 0 (d) – 1/2

14. If the pair of lines 0222 22 �� cfygxbyhxyax

intersect on the y- axis, then

(a) 222 chbgfgh �� (b) 22 chbg �

(c) fghabc 2� (d) None of these

15. The lines joining the origin to the point of intersection of

the circle 322 �� yx and the line 2�� yx are

(a) 0)223( �� xy (b) 0)223( �� yx

(c) 0)223( �� yx (d) 0)223( �� xy

Mathematics

16. The lines joining the origin to the points of intersection

of the curves 022 22 �� gxbyhxyax and

0'2''2' 22 �� xgybxyhxa wil l be mutually

perpendicular, if

(a) )(')''( bagbag �� (b) )(')''( bagbag ��

(c) )(')''( bagbag �� (d) )(')''( bagbag ��

17. Distance between the lines represented by the equation

04333332 22 �� yxyxyx is

(a) 5/2 (b) 5/4

(c) 5 (d) 0

18. If the lines joining origin to the points of intersection of

the line �� gyfx and the curve

022 �� fygxyhxyx be mutually perpendicular,,

then

(a) h�� (b) g��

(c) fg�� (d) � may have any value

19. The equation of the line joining origin to the points of

intersection of the curve 222 ayx �� and

022 �� ayaxyx is

(a) 022 �� yx (b) 0�xy

(c) 02 �� xxy (d) 02 �� xyy

20. The equation of second degree

01244222 22 �� yxyxyx represents a pair

of straight lines. The distance between them is

(a) 4 (b) 3/4

(c) 2 (d) 32

21. The equation of pair of straight lines joining the point of

intersection of the curve 422 �� yx and 2�� xy to

the origin, is

(a) 222 )( xyyx �� (b) 0)( 222 �� xyyx

(c) 222 )(4 xyyx �� (d) 0)(4 222 �� xyyx

22. The lines joining the points of intersection of line

1�� yx and curve 0222 �� yyx to the origin

are perpendicular, then the value of � will be

(a) 1/2 (b) –1/2

(c) 2/1 (d) 0

23. The lines joining the points of intersection of curve

012488125 22 �� yxyxyx and the l ine

2�� yx to the origin , makes the angles with the axes

(a) o30 and o45 (b) o45 and o60

(c) Equal (d) Parallel to axes

24. The lines joining the points of intersection of the curve

0)()( 222 �� ckyhx and the line hkhykx 2�� to

the origin are perpendicular, then

(a) khc �� (b) 222 khc ��

(c) 22 )( khc �� (d) 2224 khc ��

25. If the distance of two lines passing through origin from

the point ),( 11 yx is ''d , then the equation of lines is

(a) )()( 222211 yxdyxxy ��

(b) )()( 22211 yxxyyx ��

(c) )()( 22211 yxyxxy ��

(d) )(2)( 1122 yxyx ��

26. The equation of the locus of foot of perpendiculars drawn

from the origin to the line passing through a fixed point

(a, b), is

(a) 022 �� byaxyx (b) 022 �� byaxyx

(c) 02222 �� byaxyx

(d) None of these

27. The orthocentre of the triangle formed by the lines

0�xy and 1�� yx is

(a) )0,0( (b) ��

��

2

1,

2

1

(c) ��

��

3

1,

3

1(d) �

�

��

4

1,

4

1

28. The product of perpendiculars drawn from the origin to

the lines represented by the equation

0222 22 �� cfygxbyhxyax , will be

(a) 222 4hba

ab

��(b) 222 4hba

bc

��

(c) 222 4)( hba

ca

�� (d) 22 4)( hba

c

��

Mathematics

29. The equations to a pair of opposite sides of a

parallelogram are 0652 �� xx and 0562 �� yy .

The equations to its diagonals are

(a) 134 �� yx and 74 �� xy

(b) 134 �� yx and 74 �� xy

(c) 134 �� yx and 74 �� xy

(d) 134 �� xy and 74 �� xy

30. Area of the triangle formed by the lines

0189 22 �� xxyy and 9�y is

(a) sq4

27. units (b) .27sq units

(c) .2

27sq units (d) None of these

1.(b) 2.(c) 3.(d) 4.(c) 5.(b) 6.(d) 7.(b) 8.(c) 9.(c) 10.(d)

11.(a) 12.(d) 13.(b) 14.(a) 15.(a) 16.(b) 17.(a) 18.(d) 19.(b) 20.(c)

21.(a) 22.(d) 23.(c) 24.(b) 25.(a) 26.(a) 27.(a) 28.(d) 29.(c) 30.(a)

Mathematics

CIRCLE & SYSTEM OF CIRCLE

Definition

A circle is defined as the locus of a point which moves in a

plane such that its distance from a fixed point in that plane

always remains the same i.e., constant.

The fixed point is called the centre of the circle and the fixed

distance is called the radius of the circle.

(Moving point)

O

P

Q

Plane

Fixed point

R

Standard forms of equation of a circle

(1) General equation of a circle : The general equation of

a circle is 02222 �� cfygxyx where g, f, c are

constant.

(i) Centre of the circle is (–g, –f). i.e., (2

1� coefficient of x,

2

1� coefficient of y).

(ii) Radius of the circle is cfg �� 22 .

Nature of the circle

(i) If 022 �� cfg , then the radius of the circle will be

real. Hence, in this case, it is possible to draw a circle on

a plane.

(ii) If 022 �� cfg , then the radius of the circle will be

zero. Such a circle is known as point circle.

(iii) If 022 �� cfg , then the radius cfg �� 22 of the

circle will be an imaginary number. Hence, in this case, it

is not possible to draw a circle.

The condition for the second degree equation to

represent a circle : The general equation

22 2 byhxyax �� 022 �� cfygx represents a circle

iff

(i) 0�� ba (ii) 0�h

(iii) 02 222 �� chbgafhgfabc

(iv) 022 #�� acfg

(2) Central form of equation of a circle : The equation of

a circle having centre (h, k) and radius r is

222 )()( rkyhx ��

If the centre is origin, then the equation of the circle is

222 ryx ��

(3) Circle on a given diameter : The equation of the circle

drawn on the straight line joining two given points ),( 11 yx

and ),( 22 yx as diameter is

0))(())(( 2121 �� yyyyxxxx

2 21 1

(4) Parametric co-ordinates

(i) The parametric co-ordinates of any point on the circle

222 )()( rkyhx �� are given by )sin,cos( �� rkrh �� ,

)20( �� .

In particular, co-ordinates of any point on the circle

222 ryx �� are )sin,cos( �� rr , )20( �� .

(ii) The parametric co-ordinates of any point on the circle

02222 �� cfygxyx are

�cos)( 22 cfggx �� and

�sin)( 22 cfgfy �� , )20( ��

(5) Equation of a circle under given conditions

(i) The equation of the circle through three non-collinear

points

),(),,(),,( 332211 yxCyxByxA is0

1

1

1

1

3323

23

2222

22

1121

21

22

�

�

�

�

�

yxyx

yxyx

yxyx

yxyx

(ii) From given three points taking any two as extremities of

diameter of a circle S = 0 and equation of straight line

passing through these two points is L = 0. Then required

equation of circle is 0�� LS � , where � is a parameter,,

Mathematics

which can be found out by putting third point in the

equation.

(h,k)

P(x,y)r

C

Equation of a circle in some special cases

(1) If centre of the circle is ),( kh and it passes through origin

then its equation is 2222 )()( khkyhx �� 22 yx ��

022 �� kyhx .

(2) If the circle touches x-axis then its equation is

222 )()( kkyhx �� . (Four cases)

(3) If the circle touches y-axis then its equation is

222 )()( hkyhx �� . (Four cases)

(–h,k) (h,k)

(–h,–k) (h,–k)

h h

hh

Y

XX�

Y�

(4) If the circle touches both the axes then its equation is

222 )()( rryrx �� . (Four cases)

(–r,r) (r,r)

(–r,–r) (r,–r)

Y

XX�

Y�

(5) If the circle touches x- axis at origin then its equation is

222 )( kkyx �� 0222 �� kyyx . (Two cases)

(6) If the circle touches y-axis at origin, the equation of circle

is 222)( hyhx �� 0222 �� xhyx . (Two cases)

Y

X(–h,0) (h,0)

X�

Y�

(7) If the circle passes through origin and cut intercepts a

and b on axes, the equation of circle is

022 �� byaxyx and centre is )2/,2/( baC . (Four

cases)

Intercepts on the axes

The lengths of intercepts made by the circle

02222 �� cfygxyx on X and Y axes are cg �22

and cf �22 respectively..

Therefore,

(i) The circle 02222 �� cfygxyx cuts the x-axis in

real and distinct points, touches or does not meet in real

points according as cg �� or,2 .

(ii) Similarly, the circle 02222 �� cfygxyx cuts the

y-axis in real and distinct points, touches or does not

meet in real points according as cf �� or,2 .

Position of a point with respect to a circle

A point ),( 11 yx lies outside, on or inside a circle

02222 �� cfygxyxS according as

cfygxyxS �� 1121

211 22 is positive, zero or

negative.

The least and greatest distance of a point from a

circle: Let S = 0 be a circle and ),( 11 yxA be a point. If the

diameter of the circle through A is passing through the

circle at P and Q, then �� || rACAP least distance;

�� rACAQ greatest distance where ‘‘r’ is the radius

and C is the centre of the circle.

r C

P

Q

A

Mathematics

Intersection of a line and a circle

The length of the intercept cut off from the line

cmxy �� by the circle 222 ayx �� is

2

222

1

)1(2

m

cma

�

��.

(i) If 0)1( 222 �� cma , line will meet the circle at two real

and different points.

(ii) If )1( 222 mac �� , line will touch the circle.

(iii) If 0)1( 222 �� cma , line will meet the circle at two

imaginary points.

Tangent to a circle at a given point

(1) Point form

(i) The equation of tangent at (x1, y

1) to circle 222 ayx ��

is 211 ayyxx �� .

(ii) The equation of tangent at ),( 11 yx to circle

02222 �� cfygxyx is 0)()( 1111 �� cyyfxxgyyxx .

(2) Parametric form : Since parametric co-ordinates of a

point on the circle 222 ayx �� is ),sin,cos( �� aa then

equation of tangent at )sin,cos( �� aa is

2sin.cos. aayax �� or ayx �� sincos .

(3) Slope form : The straight line cmxy �� touches the

circle 222 ayx �� if )1( 222 mac �� and the point of

contact of tangent 21 mamxy �� is

��

�

�

��

�

�

� 22 1,

1 m

a

m

ma�

.

Length of tangent

(x1,y1)

P

Q

R

1S

Let PQ and PR be two tangents drawn from ),( 11 yxP to

the circle .02222 �� cfygxyx

Then PQ = PR is called the length of tangent drawn from

point P and is given by PQ =PR

11121

21 22 Scfygxyx �� .

Pair of tangents

From a given point ),( 11 yxP two tangents PQ and PR

can be drawn to the circle

.02222 �� cfygxyxS

Their combined equation is 21 TSS � ,

where 0�S is the equation of circle, 0�T is the

equation of tangent at ),( 11 yx and S1 is obtained by

replacing x by x1and y by y

1 in S.

Director circle

The locus of the point of intersection of two

perpendicular tangents to a circle is called the Director

circle.

Let the circle be 222 ayx �� , then equation of director

circle is 222 2ayx �� .

90°

P(x1,y1)

Obviously director circle is a concentric circle whose

radius is 2 times the radius of the given circle.

Director circle of circle 02222 �� cfygxyx is

0222 2222 �� fgcfygxyx .

Power of point with respect to a circle

Let ),( 11 yxP be a point outside the circle and PAB and

PCD drawn two secants. The power of ),( 11 yxP with

respect to 02222 �� cfygxyxS is equal to PA .

PB which is

11121

21 22 Scfygxyx ��

�� 21 )(. SPBPA

T B

A

C DP(x1,y1)

Mathematics

Square of the length of tangent.

If P is outside, inside or on the circle then PA . PB is +ve,

–ve or zero respectively.

Normal to a circle at a given point

The normal of a circle at any point is a straight line, which

is perpendicular to the tangent at the point and always

passes through the centre of the circle.

(1) Equation of normal: The equation of normal to the

circle 02222 �� cfygxyx at any point ),( 11 yx is

)( 11

11 xx

gx

fyyy �

�

�� or

fy

yy

gx

xx

�

��

�

�

1

1

1

1.

The equation of normal to the circle 222 ayx �� at any

point ),( 11 yx is 011 �� yxxy or 11 y

y

x

x� .

TangentNormal

90°

P

(2) Parametric form : Since parametric co-ordinates of a

point on the circle 222 ayx �� is )sin,cos( �� aa .

Equation of normal at )sin,cos( �� aa is

�� sincos a

y

a

x� or �� sincos

yx� or �tanxy �

or mxy � where �tan�m , which is slope form of

normal.

Chord of contact of tangents

(1) Chord of contact : The chord joining the points of

contact of the two tangents to a conic drawn from a

given point, outside it, is called the chord of contact of

tangents.

(x� ,y�) P

A(x1,y1)

(x$,y$)Q

Chord of contact

(2) Equation of chord of contact : The equation of the

chord of contact of tangents drawn from a point ),( 11 yx

to the circle 222 ayx �� is .211 ayyxx ��

Equation of chord of contact at ),( 11 yx to the circle

02222 �� cfygxyx is

0)()( 1111 �� cyyfxxgyyxx .

It is clear from above that the equation to the chord of

contact coincides with the equation of the tangent, if

point ),( 11 yx lies on the circle.

The length of chord of contact 222 pr �� ; (p being

length of perpendicular from centre to the chord)

Area of APQ� is given by 21

21

2/3221

21 )(

yx

ayxa

�

��.

(3) Equation of the chord bisected at a given point :

The equation of the chord of the circle

02222 �� cfygxyxS bisected at the point

),( 11 yx is given by 1ST � .

i.e., cfygxyxcyyfxxgyyxx �� 1121

211111 22)()( .

Common chord of two circles

(1) Definition : The chord joining the points of intersection

of two given circles is called their common chord.

(2) Equation of common chord : The equation of the

common chord of two circles

022 11122

1 �� cyfxgyxS ….(i)

and 022 22222

2 �� cyfxgyxS ….(ii)

is 0)(2)(2 212121 �� ccffyggx i.e., 021 �� SS .

(3) Length of the common chord :

21

212)(2 MCPCPMPQ ��

Where �PC1 radius of the circle 01 �S and �MC1

length of the perpendicular from the centre 1C to the

common chord PQ.

Diameter of a circle

The locus of the middle points of a system of parallel

chords of a circle is called a diameter of the circle.

The equation of the diameter bisecting parallel chords

cmxy �� (c is a parameter) of the circle 222 ayx �� is

.0��myx

Common tangents to two circles

Different cases of intersection of two circles :

Mathematics

Let the two circles be 21

21

21 )()( ryyxx �� …..(i)

and 22

22

22 )()( ryyxx �� …..(ii)

with centres ),( 111 yxC and ),( 222 yxC and radii r1 and r

2

respectively. Then following cases may arise :

Case I : When 2121 || rrCC �� i.e., the distance between

the centres is greater than the sum of radii.

C1

r2

C2 T

r1

P

Direct common

Transverse common

In this case four common tangents can be drawn to the

two circles, in which two are direct common tangents

and the other two are transverse common tangents.

The points P, T of intersection of direct common tangents

and transverse common tangents respectively, always

lie on the line joining the centres of the two circles and

divide it externally and internally respectively in the ratio

of their radii.

2

1

2

1

r

r

PC

PC� (externally) and

2

1

2

1

r

r

TC

TC� (internally)

Hence, the ordinates of P and T are

��

��

�

�

�

��

21

1221

21

1221 ,rr

yryr

rr

xrxrP and ��

�

��

�

�

�

��

21

1221

21

1221 ,rr

yryr

rr

xrxrT .

Case II : When 2121 || rrCC �� i.e., the distance between

the centres is equal to the sum of radii.

Direct common tangents

Transverse common tangent

C2 C1 T P

In this case two direct common tangents are real and

distinct while the transverse tangents are coincident.

Case III : When 2121 || rrCC �� i.e., the distance between

the centres is less than sum of radii.Direct common

tangents

PC2 C1

In this case two direct common tangents are real and

distinct while the transverse tangents are imaginary.

Case IV : When ,|||| 2121 rrCC �� i.e., the distance

between the centres is equal to the difference of the

radii.

1

2

2

1

In this case two tangents are real and coincident while

the other two tangents are imaginary.

Case V : When ,|||| 2121 rrCC �� i.e., the distance

between the centres is less than the difference of the

radii.

C1

r2

C2

r1

In this case, all the four common tangents are imaginary.

Angle of intersection of two circles

The angle of intersection between two circles S = 0 and

S’ = 0 is defined as the angle between their tangents at

their point of intersection.

If 022 11122 �� cyfxgyxS

022' 22222 �� cyfxgyxS

S=0

�–�r1

C1

B AQ

C2

r2�

S�=0

P

A� B�

are two circles with radii 21, rr and d be the distance

between their centres then the angle of intersection �

between them is given by 21

222

21

2cos

rr

drr �� or

22

2221

21

21

212121

2

)()(2cos

cfgcfg

ccffgg

��

��

.

Condition of Orthogonality : If the angle of

intersection of the two circles is a right angle )90( o�� ,

then such circles are called orthogonal circles and

condition for orthogonality is 212121 22 ccffgg �� .

C1 C2

90°

P

(–g1,–f1) (–g2,–f2)

Family of circles

(1) The equation of the family of circles passing through the

Mathematics

point of intersection of two given circles S = 0 and S’ = 0

is given as 0'�� SS � ,

(where � is a parameter, )1��

S=0 S�=0

S+�S�=0(2) The equation of the family of circles passing through the

point of intersection of circle S = 0 and a line L = 0 is

given as 0�� LS � , (where � is a parameter)

S=0 S+�L=0L=0

(3) The equation of the family of circles touching the circle

S = 0 and the line L = 0 at their point of contact P is

0�� LS � , (where � is a parameter)

S=0S+�L=0L=0

(4) The equation of a family of circles passing through two

given points ),( 11 yxP and ),( 22 yxQ can be written in

the form

0

1

1

1

)()()()(

22

112121 ��

yx

yx

yx

yyyyxxxx �

,(where � is a parameter)

P(x1,y1)

Q(x2,y2)

(5) The equation of family of circles, which touch

)( 11 xxmyy �� at ),( 11 yx for any finite m is

){()()( 12

12

1 yyyyxx �� 0)}( 1 �� xxm

And if m is infinite, the family of circles is

0)()()( 12

12

1 �� xxyyxx � ,

(where � is a parameter)

(6) Equation of the circles given in diagram is

�� )()( 21 xxxx )(){(cot)()( 2121 yyxxyyyy ��

(x2,y2)

�

�

(x1,y1)

1. The straight line 0)3()2( �� yx cuts the circle

11)3()2( 22 �� yx at

(a) No points (b) One point

(c) Two points (d) None of these

2. A circle lies in the second quadrant and touches both

the axes. If the radius of the circle be 4, then its equation

is

(a) 0168822 �� yxyx

(b) 0168822 �� yxyx

(c) 0168822 �� yxyx

(d) 0168822 �� yxyx

3. The equation of the circle whose centre is (3, –1) and

which cuts off a chord of length 6 on the line

01852 �� yx is

(a) 38)1()3( 22 �� yx

(b) 38)1()3( 22 �� yx

(c) 38)1()3( 22 �� yx

(d) None of these

4. A circle has its equation in the form

014222 �� yxyx . Choose the correct

coordinates of its centre and the right value of its radius

from the following

(a) Centre (–1, –2), radius = 2

(b) Centre (2, 1), radius = 1

(c) Centre (1, 2), radius = 3

(d) Centre (–1, 2), radius = 2

5. If the point (2, 0), (0, 1), (4, 5) and (0, c) are con-cyclic,then c is equal to

(a)14

3,1 �� (b)

3

14,1 ��

(c) 1,3

14(d) None of these

Mathematics

6. The point )7,10(P lies outside the circle

0202422 �� yxyx . The greatest distance of P

from the circle is

(a) 5 (b) 3

(c) 5 (d) 15

7. The diameter of a circle is AB and C is another point on

circle, then the area of triangle ABC will be

(a) Maximum, if the triangle is isosceles

(b) Minimum, if the triangle is isosceles

(c) Maximum, if the triangle is equilateral

(d) None of these

8. If a circle of constant radius 3k passes through the origin

and meets the axes at A and B, the locus of the centroid

of the triangle OAB is the circle

(a) 222 kyx �� (b) 0222 �� kyx

(c) 04 222 �� kyx (d) 222 4kyx ��

9. The equation of the image of the circle

0183241622 �� yxyx by the line mirror

01374 �� yx is

(a) 023543222 �� yxyx

(b) 023543222 �� yxyx

(c) 023543222 �� yxyx

(d) 023543222 �� yxyx

10. Locus of a point which moves such that sum of the

squares of its distances from the sides of a square of side

unity is 9, is

(a) Straight line (b) Circle

(c) Parabola (d) None of these

11. ABCD is a square, the length of whose side is a. Taking AB

and AD as the coordinate axes, the equation of the circle

passing through the vertices of the square, is

(a) 022 �� ayaxyx (b) 022 �� ayaxyx

(c) 02222 �� ayaxyx

(d) 02222 �� ayaxyx

12. Locus of the point given by the equations 21

2

t

atx

�� ,

)11(1

)1(2

2

��

�� t

t

tay is a

(a) Straight line (b) Circle

(c) Ellipse (d) Hyperbola

13. The equation of the circle with origin as centre passing

the vertices of an equilateral triangle whose median is

of length 3a is

(a) 222 9ayx �� (b) 222 16ayx ��

(c) 222 ayx �� (d) None of these

14. If the line 0143 �� yx touches the circle222 )2()1( ryx �� , then the value of r will be

(a) 2 (b) 5

(c)5

12(d)

5

2

15. The two points A and B in a plane are such that for all

points P lies on circle satisfied kPB

PA� , then k will not

be equal to

(a) 0 (b) 1

(c) 2 (d) None of these

16. The locus of a point which divides the join of )1,1(�A

and a variable point P on the circle 422 �� yx in the

ratio 3 : 2, is

(a) 028)(20)(25 22 �� yxyx

(b) 028)(20)(25 22 �� yxyx

(c) 028)(25)(20 22 �� yxyx

(d) None of these

17. The abscissae of A and B are the roots of the equation

02 22 �� baxx and their ordinates are the roots of

the equation 02 22 �� qpyy . The equation of the

circle with AB as diameter

(a) 022 2222 �� qbpyaxyx

(b) 02 2222 �� qbpyaxyx

(c) 022 2222 �� qbpyaxyx

(d) None of these

18. A square is inscribed in the circle

0934222 �� yxyx with its sides parallel to the

coordinate axes. The coordinates of its vertices are

(a) (–6, –9), (–6, 5), (8, –9) and (8, 5)

(b) (–6, 9), (–6, –5), (8, –9) and (8, 5)

(c) (–6, –9), (–6, 5), (8, 9) and (8, 5)

(d) (–6, –9), (–6, 5), (8, –9) and (8, –5)

Mathematics

19. Chord of contact of the point (3, 2) w.r.t. the circle

2522 �� yx meets the coordinate axes in A and B. The

circumcentre of triangle OAB is

(a) ��

��

6

25,

4

25(b) �

�

��

50

3,

50

2

(c) ��

��

4

25,

6

25(d) None of these

20. The circle 422 �� yx cuts the line joining the points

)0,1(A and )4,3(B in two points P and Q. Let ��PA

BP

and ��QA

BQ. Then � and � are roots of the quadratic

equation

(a) 02123 2 �� xx (b) 02123 2 �� xx

(c) 02132 2 �� xx (d) None of these

21. A circle is inscribed in an equilateral triangle of side a,

the area of any square inscribed in the circle is

(a)3

2a(b)

3

2 2a

(c)6

2a(d)

12

2a

22. Let 1L be a straight line passing through the origin and

2L be the straight line 1�� yx . If the intercepts made

by the circle 0322 �� yxyx on 1L and 2L are

equal, then which of the following equations can

represent 1L

(a) 0�� yx (b) 0�� yx

(c) 07 �� yx (d) 07 �� yx

23. The area of the triangle formed by joining the origin to

the points of intersection of the line

5325 �� yx and circle 1022 �� yx is

(a) 3 (b) 4

(c) 5 (d) 6

24. The centre of the circle passing through (0, 0) and (1, 0)

and touching the circle 922 �� yx is

(a) ��

��

2

1,

2

1(b) �

�

��

� 2,

2

1

(c) ��

��

2

1,

2

3(d) �

�

��

2

3,

2

1

25. If 4,3,2,1,1

, ��

��

i

mm

ii are con-cyclic points, then the

value of 4321 ... mmmm is

(a) 1 (b) – 1


26. The normal at the point (3, 4) on a circle cuts the circle atthe point (–1, –2). Then the equation of the circle is

(a) 0132222 �� yxyx

(b) 0112222 �� yxyx

(c) 0122222 �� yxyx

(d) 0142222 �� yxyx

27. Tangents are drawn from the point (4, 3) to the circle

922 �� yx . The area of the triangle formed by them

and the line joining their points of contact is

(a)25

24(b)

25

64

(c)25

192(d)

5

192

28. If the tangent to the circle 222 ryx �� at the point (a, b)

meets the coordinate axes at the point A and B, and O is

the origin, then the area of the triangle OAB is

(a)ab

r

2

4

(b)ab

r4

(c)ab

r

2

2

(d)ab

r2

29. The co-ordinates of the point from where the tangents

are drawn to the circles 122 �� yx ,

015822 �� xyx and 0241022 �� yyx are of

same length, are

(a) ��

��

2

5,2 (b) �

�

��

��

2

5,2

(c) ��

��

�

2

5,2 (d) �

�

��

�

2

5,2

30. The tangents are drawn from the point (4, 5) to the circle

0112422 �� yxyx . The area of quadrilateral

formed by these tangents and radii, is

(a) 15 sq. units (b) 75 sq. units

(c) 8 sq. units (d) 4 sq. units

Mathematics

31. Let PQ and RS be tangents at the extremeties of the

diameter PR of a circle of radius r. If PS and RQ intersect

at a point X on the circumference of the circle, then 2r

equals

(a) RSPQ. (b)2

RSPQ �

(c)RSPQ

RSPQ

�

.2(d)

2

22 RSPQ �

32. The angle between a pair of tangents drawn from a point

P to the circle 96422 �� yxyx

0cos13sin 22 �� is �2 . The equation of the locus

of the point P is

(a) 046422 �� yxyx

(b) 096422 �� yxyx

(c) 046422 �� yxyx

(d) 096422 �� yxyx

33. If a straight line through )8,8(�C making an angle of

%135 with the x-axis cuts the circle �� sin5,cos5 �� yx

at points A and B, then the length of AB is

(a) 3 (b) 7


34. The number of common tangents to the circles

422 �� yx and 248622 �� yxyx is

(a) 0 (b) 1

(c) 3 (d) 4

37. If two distinct chords, drawn from the point (p, q) on

the circle qypxyx �� 22 , (where 0�pq ) are

bisected by the x-axis, then

(a) 22 qp � (b) 22 8qp �

(c) 22 8qp � (d) 22 8qp �

38. From the origin, chords are drawn to the circle

1)1( 22 �� yx . The locus of mid points of these chords

is a

(a) Circle (b) Straight line

(c) Pair of straight line (d) None of these

39. Let AB be a chord of the circle 222 ryx �� subtending a

right angle at the centre. Then the locus of the centroid

of the PAB� as P moves on the circle is

(a) A parabola (b) A circle

(c) An ellipse (d) A pair of straight lines

40. If the circle 1C : 1622 �� yx intersects another circle

2C of radius 5 in such a manner that the common chord

is of maximum length and has a slope equal to 4

3, the

coordinates of the centre of 2C are

(a) ��

��

�

5

12,

5

9, �

�

��

�

5

12,

5

9(b) �

�

��

��

��

��

5

12,

5

9,

5

12,

5

9

(c) ��

��

��

��

�

5

12,

5

9,

5

12,

5

9(d) None of these

1. (a) 2. (b) 3. (a) 4 . (a) 5. (c) 6. (d) 7. (a) 8. (d) 9. (d) 10. (b)

11. (b) 12. (b) 13. (d) 14. (a) 15. (b) 16. (a) 17. (a) 18. (a) 19. (d) 20. (a)

21. (c) 22. (bc) 23. (c) 24. (b) 25. (a) 26. (b) 27. (c) 28. (a) 29. (b) 30. (c)

31. (a) 32. (d) 33. (c) 34. (b) 35. (a) 36. (b) 37. (d) 38. (a) 39. (b) 40. (a)

Mathematics

CONIC SECTION

Definition

The curves obtained by intersection of a plane and a double

cone in different orientation are called conic section.

Definitions of various important terms

(1) Focus : The fixed point is called the focus of the conic-

section.

(2) Directrix : The fixed straight line is called the directrix of

the conic section.

(3) Eccentricity : The constant ratio is called the eccentricity

of the conic section and is denoted by e.

(4) Axis : The straight line passing through the focus and

perpendicular to the directrix is called the axis of the

conic section. A conic is always symmetric about its axis.

(5) Vertex : The points of intersection of the conic section

and the axis are called vertices of conic section.

(6) Centre : The point which bisects every chord of the

conic passing through it, is called the centre of conic.

(7) Latus-rectum : The latus-rectum of a conic is the chord

passing through the focus and perpendicular to the axis.

(8) Double ordinate : The double ordinate of a conic is a

chord perpendicular to the axis.

(9) Focal chord : A chord passing through the focus of the

conic is called a focal chord.

(10) Focal distance : The distance of any point on the conic

from the focus is called the focal distance of the point.

General equation of a conic section when its focus,

directrix and eccentricity are given

Let ),( ��S be the focus, 0�� CByAx be the directrix

and e be the eccentricity of a conic. Let ),( khP be any

point on the conic. Let PM be the perpendicular from P,

on the directrix. Then by definition,

),( ��S

P(h, k)

Z

M

Z�

ePMSP � � 222 PMeSP �

�

2

22

222 )()(��

�

�

��

�

��

BA

CBkAhekh ��

Thus the locus of ),( kh is

�� 22 )()( �� yx )(

)(22

22

'�

��

A

CByAxe ,

which is general equation of second degree.

Recognisation of conics

The equation of conics is represented by the general

equation of second degree

0222 22 �� cfygxbyhxyax ......(i)

and discriminant of above equation is represented by

� , where 2222 chbgaffghabc ��

Case I : When 0�� .

In this case equation (i) represents the degenerate conic

whose nature is given in the following table.

S. No. Condition Nature of conic 1. 0�� and `02 �� hab A pair of coincident

straight lines 2. 0�� and 02 �� hab A pair of intersecting

straight lines 3. 0�� and 02 �� hab A point

Case II : When 0�� .

In this case equation (i) represents the non-degenerate

conic whose nature is given in the following table.S. No. Condition Nature of conic

1. bah �� ,0,0 , e = 0 A circle

2. 0,0 2 �� hab , e = 1 A parabola

3. 0,0 2 �� hab , e < 1 An ellipse

4. 0,0 2 �� hab , e >1 A hyperbola

5. 0,0 2 �� hab , 0�� ba , 2�e A rectangular hyperbola

PARABOLA

Definition

A parabola is the locus of a point which moves in a plane

such that its distance from a fixed point (i.e., focus) in

the plane is always equal to its distance from a fixed

straight line (i.e., directrix) in the same plane.

Standard equation of the parabola

Let S be the focus, 'ZZ be the directrix of the parabola

and ),( yx be any point on parabola, then standard form

of the parabola is axy 42 � .

Some other standard forms of parabola are

(i) Parabola opening to left i.e, axy 42 ��

Mathematics

(ii) Parabola opening upwards i.e., ayx 42 �

(iii) Parabola opening downwards i.e., ayx 42 ��

Some terms related to parabola

Important terms

axy2 4� axy2 4�� ayx 2 4� ayx 2 4��

Vertex (0, 0) (0, 0) (0, 0) (0, 0)

Focus (a, 0) (–a, 0) (0, a) (0, –a)

Directrix ax �� ax � ay �� y = a

Axis 0�y 0�y 0�x 0�x

Latusrectum a4 a4 a4 a4

Focal distance

),( yxP

ax � xa � ay � ya �

Special form of parabola (y – k)2 = 4a(x – h) = a

The equation of a parabola with its vertex at (h, k) and

axis as parallel to x-axis is )(4)( 2 hxaky �� .

If the vertex of the parabola is ),( qp and its axis is parallel

to y-axis, then the equation of the parabola is

)(4)( 2 qybpx �� .

Parametric equations of a parabola

Parabola axy 42 � axy 42 �� ayx 42 � ayx 42 ��

Parametric

Co-ordinates )2,( 2 atat )2,( 2 atat� ),2( 2atat ( 2,2 atat � )

Parametric

Equations

2atx �

aty 2�

2atx ��

aty 2�

atx 2�2aty �

atx 2� , 2aty ��

Position of a point and a line with respect to a parabola

(1) Position of a point with respect to a parabola : The

point ),( 11 yxP lies outside, on or inside the parabola

axy 42 � according as 0,,4 121 �� axy .

X

Y

P(inside)P

(Outside)

P(on)

(2) Intersection of a line and a parabola: The line

cmxy �� does not intersect, touches or intersect a

parabola axy 42 � , according as m

ac �� ,, .

Condition of tangency : The line cmxy �� touches

the parabola, if mac /� .

Equations of tangent in different forms

(1) Point Form

Equations of tangent of all other standard parabolas at (x1, y1)

Equation of parabola Tangent at (x1, y1)

y2 = 4ax yy1 = 2a (x + x1)

axy 42 �� )(2 11 xxayy ��

ayx 42 � )(2 11 yyaxx ��

ayx 42 �� )(2 11 yyaxx ��

(2) Parametric form

Equations of tangent of all other standard parabolas at 't'

Equations of parabolas

Parametric co-ordinates 't'

Tangent at 't'

axy 42 � atat 2,2 2atxty ��

axy 42 �� )2,( 2 atat� 2atxty ��

ayx 42 � ),2( 2atat 2atytx ��

ayx 42 �� ),2( 2atat � 2atytx ��

(3) Slope Form

Equations of tangent of all other parabolas in slope form

Equation of

parabolas

Point of contact in

terms of slope (m)

Equation of tangent in

terms of slope (m)

Condition of

Tangency

axy 42 ��

��

m

a

m

a 2,

2 m

amxy ��

m

ac �

axy 42 ��

��

��m

a

m

a 2,

2 m

amxy ��

m

ac ��

ayx 42 � ),2( 2amam 2ammxy �� 2amc ��

ayx 42 �� ),2( 2amam �� 2ammxy �� 2amc �

Point of intersection of tangents at any two points on

the parabola

(1) The point of intersection of tangents at two points

)2,( 121 atatP and )2,( 2

22 atatQ on the parabola axy 42 � is

))(,( 2121 ttatat � .

Mathematics

222

1 2 1 2

121

(2) The locus of the point of intersection of tangents to the

parabola axy 42 � which meet at an angle � is

axyax 4tan)( 222 �� .

(3) Director circle: The locus of the point of intersection of

perpendicular tangents to a conic is known as its director

circle. The director circle of a parabola is its directrix.

(4) The tangents to the parabola axy 42 � at )2,( 121 atatP

and )2,( 222 atatQ intersect at R. Then the area of triangle

PQR is 3

212 )(

2

1tta � .

Equation of pair of tangents from a point to a parabola

The combined equation of the pair of the tangents drawn

from a point to a parabola is 2' TSS � , where

;42 axyS �� 121 4' axyS �� and )(2 11 xxayyT �� .

1 1

The two tangents can be drawn from a point to a

parabola. The two tangent are real and distinct or

coincident or imaginary according as the given point

lies outside, on or inside the parabola.

Equations of normal in different forms

(1) Point form

Equation of normals of all other standard parabolas at (x1, y1)

Equation of parabola Normal at (x1, y1)

y2 = 4ax

y – y1 = a

y

2

1�

(x – x1)

axy 42 �� )(2

11

1 xxa

yyy ��

ayx 42 � )(2

1

1

1 xxx

ayy ��

ayx 42 �� )(2

1

1

1 xxx

ayy ��

(2) Parametric form

Equations of normal of all other standard parabola at 't'

Equations of parabolas

Parametric co-ordinates

Normals at 't

y2 = 4ax (at2, 2at) y + tx = 2at + at3

axy 42 �� )2,( 2 atat� 32 atattxy ��

ayx 42 � ),2( 2atat 32 atattyx ��

ayx 42 �� ),2( 2atat � 32 atattyx ��

(3) Slope formEquations of normal, point of contact, and condition of

normality in terms of slope (m)

Equations of

parabola

Point of contact in terms of slope (m)

Equations of normal in terms

of slope (m)

Condition of

normality

axy 42 � )2,( 2 amam � 32 amammxy �� 32 amamc ��

axy 42 �� )2,( 2 amam� 32 amammxy �� 32 amamc ��

ayx 42 � �

�

��

�

2,

2

m

a

m

a2

2m

aamxy ��

22

m

aac ��

ayx 42 ��

��

�

2,

2

m

a

m

a2

2m

aamxy ��

22

m

aac ��

Point of intersection of normals at any two points on

the parabola

The point of intersection of normals at any two points

)2,( 121 atatP and )2,( 2

22 atatQ on the parabola axy 42 � is

)](),(2[ 21212122

21 tttatttttaaR ��

121

222

Relation between ‘t1’ and ‘t

2’ if normal at ‘t

1’ meets the

parabola again at ‘t2’

If the normal at the point )2,( 121 atatP meets the

parabola axy 42 � again at )2,( 222 atat ,

then 1

12

2

ttt �� .

)2,( 121 atatY

XA

Y�)2,( 2

22 atat

Q

P

Mathematics

Co-normal points

The points on the curve at which the normals pass

through a common point are called co-normal points. Q,

R, S are co-normal points. The co- normal points are also

called the feet of the normals.

Y

X� X

RO

Y�

P(x1,y1)

Q

S

Properties of co-normal points

(1) Three normals can be drawn from a point to a parabola.

(2) The algebraic sum of the slopes of three concurrent

normals is zero.

(3) The sum of the ordinates of the co-normal points is zero.

(4) The centroid of the triangle formed by the co-normal

points lies on the axis of the parabola.

(5) The centroid of a triangle formed by joining the foots of

the normal of the parabola lies on its axis and is given by

��

��

��

3

222,

3321

23

22

21 amamamamamam

= ��

��

��0,

3

23

22

21 amamam

.

(6) If three normals drawn to any parabola axy 42 � from a

given point (h, k) be real, then ah 2� for 1�a , normals

drawn to the parabola xy 42 � from any point (h, k) are

real, if 2�h .

(7) Out of these three at least one is real, as imaginary

normals will always occur in pairs.

Equation of the chord of contact of tangents to a

parabola

Let PQ and PR be tangents to the parabola axy 42 �

drawn from any external point ),( 11 yxP then QR is called

the chord of contact of the parabola axy 42 � .

(x1,y1)PO

Y

X�

Q

R

Chord of contact

Y'

X�

The chord of contact of tangents drawn from a point

),( 11 yx to the parabola axy 42 � is )(2 11 xxayy �� .

Equation of the chord of the parabola which is bisected

at a given point

The equation of the chord at the parabola axy 42 �

bisected at the point ),( 11 yx is given by ,1ST �

i.e., )(2 11 xxayy �� 121 4axy ��

where )(2 11 xxayyT �� and 1211 4axyS �� .

Equation of the chord joining any two points on the

parabola

Let )2,(),2,( 22,21

21 atatQatatP be any two points on the

parabola axy 42 � . Then, the equation of the chordd

joining these points is, 21

211

22 atx

ttaty �

�� or

2121 22)( tatxtty �� .

(1) Condition for the chord joining points having

parameters t1 and t

2 to be a focal chord: If the chord

joining points )2,( 121 atat and )2,( 2

22 atat on the parabola

passes through its focus, then )0,(a satisfies the equation

2121 22)( tatxtty ��

� 21220 tata �� 121 ��tt or 1

2

1

tt �� .

(2) Length of the focal chord: The length of a focal chord

having parameters 1t and 2t for its end points is

212 )( tta � .

ELLIPSE

Definition

An ellipse is the locus of a point which moves in such a

way that its distance from a fixed point is in constant

ratio (<1) to its distance from a fixed line. The fixed point

is called the focus and fixed line is called the directrix

and the constant ratio is called the eccentricity of the

ellipse, denoted by (e).

Standard equation of the ellipse

Let S be the focus, ZM be the directrix of the ellipse and

),( yxP is any point on the ellipse, then by definition

12

2

2

2

��b

y

a

x, where )1( 222 eab �� .

Mathematics

Since 1�e , therefore 222 )1( aea �� 22 ab � .

The other form of equation of ellipse is 12

2

2

2

��b

y

y

x,

where, baeieba �� .,.)1( 222 .

Difference between both ellipses will be clear from the

following table :

Ellipse

Imp.

terms ()

(*+

(,

(-.

�� 12

2

2

2

b

y

a

x

For a > b For b > a

Centre (0, 0) (0, 0)

Vertices )0,( a� ),0( b�

Length of major axis 2a 2b

Length of minor axis 2b 2a

Foci )0,( ae� ),0( be�

Equation of directrices

eax /�� eby /��

Relation in a, b and e )1( 222 eab �� )1( 222 eba ��

Length of latus rectum

a

b22

b

a22

Ends of latus-rectum

��

��

��a

bae

2

, ��

��

�� be

b

a,

2

Parametric equations )sin,cos( �� ba )sin,cos( �� ba

)20( ��

Focal radii 1exaSP ��

1' exaPS ��

1eybSP ��

1' eybPS ��

Sum of focal radii

�� PSSP '

2a 2b

Distance between foci 2ae 2be

Distance between directrices

2a/e 2b/e

Tangents at the vertices

x = –a, x = a y = b, y = –b

Parametric form of the ellipse

Let the equation of ellipse in standard form will be given

by 12

2

2

2

��b

y

a

x.

Then the equation of ellipse in the parametric form will

be given by �� sin,cos byax �� , where � is the

eccentric angle whose value vary from �� 20 �� .

Therefore coordinate of any point P on the ellipse will

be given by )sin,cos( �� ba .

Special forms of an ellipse

(1) If the centre of the ellipse is at point ),( kh and the

directions of the axes are parallel to the coordinate axes,

then its equation is 1)()(

2

2

2

2

��

��

b

ky

a

hx.

(2) If the equation of the curve is 2

2)(

a

nmylx ��

1)(

2

2

��

�b

plymx, where 0�� nmylx and

0�� plymx are perpendicular lines, then we

substitute ,22

Xml

nmylx�

�

��Y

ml

plymx�

�

��22 , to put the

equation in the standard form.

Position of a point with respect to an ellipse

Let ),( 11 yxP be any point and let 12

2

2

2

��b

y

a

x is the

equation of an ellipse. The point lies outside, on or inside

the ellipse as if 0,,12

21

2

21

1 ��b

y

a

xS

Intersection of a line and an ellipse

The line cmxy �� intersects the ellipse 12

2

2

2

��b

y

a

x

in two distinct points if 2222 cbma �� , in one point if

2222 bmac �� and does not intersect if

2222 cbma �� .

Mathematics


(x1,y1)P

A

B

(1) Point form: The equation of the tangent to the ellipse

12

2

2

2

��b

y

a

x at the point ),( 11 yx is 1

21

21 ��

b

yy

a

xx.

(2) Slope form: If the line cmxy �� touches the ellipse

122

2

��

b

y

a

x, then 2222 bmac �� . Hence, the straight

line 222 bmamxy �� always represents the tangents

to the ellipse.

Points of contact: Line 222 bmamxy �� touches

the ellipse 122

2

��

b

y

a

x at �

�

�

�

��

��

�222

2

222

2

,bma

b

bma

ma �

.

(3) Parametric form: The equation of tangent at any point

)sin,cos( �� ba is 1sincos �� b

y

a

x.

Equation of pair of tangents SS1 = T2

Pair of tangents: The equation of pair of tangents PA

and PB is 21 TSS � ,

where 12

2

2

2

��b

y

a

xS

12

21

2

21

1 ��b

y

a

xS

12

1

2

1 ��b

yy

a

xxT

Director circle: The director circle is the locus of points

from which perpendicular tangents are drawn to the

ellipse.

Hence locus of ),( 11 yxP i.e., equation of director circle is

2222 bayx �� .


(1) Point form: The equation of the normal at ),( 11 yx too

the ellipse 12

2

2

2

��b

y

a

xis

22

1

2

1

2

bay

yb

x

xa�� .

(2) Parametric form: The equation of the normal to the

ellipse 12

2

2

2

��b

y

a

x at )sin,cos( �� ba is

�� eccossec byax 22 ba � .

(3) Slope form: If m is the slope of the normal to the ellipse

12

2

2

2

��b

y

a

x, then the equation of normal is

222

22 )(

mba

bammxy

�

�� .

The co-ordinates of the point of contact are

��

�

�

��

�

�

�

�222

2

222

2

,mba

mb

mba

a .

Auxiliary circle

The circle described on the major axis of an ellipse as

diameter is called an auxiliary circle of the ellipse.

If 12

2

2

2

��b

y

a

x is an ellipse, then its auxiliary circle is

222 ayx �� .

Eccentric angle of a point: Let P be any point on the

ellipse 12

2

2

2

��b

y

a

x. Draw PM perpendicular from P on

the major axis of the ellipse and produce MP to meet

the auxiliary circle in Q. Join CQ. The angle ��XCQ is

called the eccentric angle of the point P on the ellipse.

Note that the angle XCP� is not the eccentric angle of

point P.

Chord of contact

1 1

If PQ and PR be the tangents through point ),( 11 yxP to

the ellipse ,12

2

2

2

��b

y

a

x then the equation of the chord

of contact QR is 121

21 ��

b

yy

a

xx or 0�T at ),( 11 yx .

Equation of chord with mid point (x1, y

1)

Mathematics

The equation of the chord of the ellipse

,12

2

2

2

��b

y

a

xwhose mid point be ),( 11 yx is 1ST �

where 121

21 ��

b

yy

a

xxT ,

12

21

2

21

1 ��b

y

a

xS .

Equation of the chord joining two points on an ellipse

The equation of the chord joining two points having

eccentric angles � and � on the ellipse 12

2

2

2

��b

y

a

x is

��

��

��

�

��

��

�

��

�

2cos

2sin

2cos

��b

y

a

x.

HYPERBOLA

Definition

A hyperbola is the locus of a point in a plane which

moves in the plane in such a way that the ratio of its

distance from a fixed point in the same plane to its

distance from a fixed line is always constant which is

always greater than unity.

Standard equation of the hyperbola

Let S be the focus, ZM be the directrix and e be the

eccentricity of the hyperbola, then by definition,

12

2

2

2

��b

y

a

x, where )1( 222 �� eab .

Conjugate hyperbola

The hyperbola whose transverse and conjugate axis are

respectively the conjugate and transverse axis of a given

hyperbola is called conjugate hyperbola of the given

hyperbola.

Difference between both hyperbolas will be clear from

the following table :

Hyperbola

Imp. terms

12

2

2

2

��b

y

a

x1

2

2

2

2

��b

y

a

x or

12

2

2

2

��b

y

a

x

Centre (0, 0) (0, 0) Length of transverse axis

2a 2b

Length of conjugate axis

2b 2a

Foci )0,( ae� ),0( be�

Equation of directrices eax /�� eby /��

Eccentricity

��

��

��

2

22

a

bae �

��

��

��

2

22

b

bae

Length of latus rectum ab /2 2 ba /2 2

Parametric co-ordinates

)tan,sec( �� ba

�� 20 ��

)tan,sec( �� ab

�� 20 ��

Focal radii aexSP �� 1

aexPS ��1

beySP �� 1

beyPS ��1

Difference of focal

radii )( SPPS ��2a 2b

Tangents at the vertices

axax �� , byby �� ,

Equation of the transverse axis

0�y 0�x

Equation of the conjugate axis

0�x 0�y

Special form of hyperbola

If the centre of hyperbola is (h, k) and axes are parallel to

the co-ordinate axes, then its equation is

1)()(

2

2

2

2

��

��

b

ky

a

hx.

Auxiliary circle of hyperbola

Let 12

2

2

2

��b

y

a

x be the hyperbola, then equation of the

auxiliary circle is 222 ayx �� .

Y�

Y

X(– a,0)A� (0,0)C

Q

90o

N

P(x,y)

X�A(a,0)

�

Let ��QCN . Here P and Q are the corresponding

points on the hyperbola and the auxiliary circle

)20( �� .

Parametric equations of hyperbola

The equations �secax � and �tanby � are known as

the parametric equations of the hyperbola 12

2

2

2

��b

y

a

x.

This )tan,sec( �� ba lies on the hyperbola for all values

of � .

Mathematics

Position of a point with respect to a hyperbola

Let the hyperbola be 12

2

2

2

��b

y

a

x.

Then ),( 11 yxP will lie inside, on or outside the hyperbola

12

2

2

2

��b

y

a

x according as 1

2

21

2

21 ��

b

y

a

x is positive, zero

or negative.

Intersection of a line and a hyperbola

The straight line cmxy �� will cut the hyperbola

12

2

2

2

��b

y

a

x in two points may be real, coincident or

imaginary according as 2222 ,, bmac �� .

Condition of tangency : If straight line cmxy ��

touches the hyperbola 12

2

2

2

��b

y

a

x, then

2222 bmac �� .


(1) Point form : The equation of the tangent to the

hyperbola 12

2

2

2

��b

y

a

x at ),( 11 yx is 1

21

21 ��

b

yy

a

xx.

(2) Parametric form : The equation of tangent to the

hyperbola 12

2

2

2

��b

y

a

x at )tan,sec( �� ba is

1tansec �� b

y

a

x.

(3) Slope form : The equations of tangents of slope m to

the hyperbola 12

2

2

2

��b

y

a

x are 222 bmamxy �� and

the co-ordinates of points of contacts are

��

�

�

��

��

��

222

2

222

2

,bma

b

bma

ma.

Equation of pair of tangents

If ),( 11 yxP be any point outside the hyperbola

12

2

2

2

��b

y

a

x then a pair of tangents PQ, PR can be drawn

to it from P.

1 1

The equation of pair of tangents PQ and PR is 21 TSS �

where, 12

2

2

2

��b

y

a

xS , 1,1

21

21

2

21

2

21

1 ��b

yy

a

xxT

b

y

a

xS

Director circle : The director circle is the locus of points

from which perpendicular tangents are drawn to the

given hyperbola. The equation of the director circle of

the hyperbola 12

2

2

2

��b

y

a

x is 2222 bayx �� .


(1) Point form : The equation of normal to the hyperbola

12

2

2

2

��b

y

a

x at ),( 11 yx is

22

1

2

1

2

bay

yb

x

xa�� .

(2) Parametric form: The equation of normal at

)tan,sec( �� ba to the hyperbola 12

2

2

2

��b

y

a

x is

�� cotcos byax � = 22 ba �

(3) Slope form: The equation of the normal to the hyperbola

12

2

2

2

��b

y

a

x in terms of the slope m of the normal is

222

22 )(

mba

bammxy

�

�� .

(4) Condition for normality : If cmxy �� is the normal

of 12

2

2

2

��b

y

a

x, then 222

22 )(

bma

bamc

�

�� or

)(

)(222

22222

bma

bamc

�

�� , which is condition of normality..

(5) Points of contact : Co-ordinates of points of contact

are ��

�

�

��

��

222

2

222

2

,mba

mb

mba

a� .

Equation of chord of contact of tangents drawn from a

point to a hyperbola

Let PQ and PR be tangents to the hyperbola

12

2

2

2

��b

y

a

x drawn from any external point ),( 11 yxP .

Mathematics

Then equation of chord of contact QR is 12

1

2

1 ��b

yy

a

xx

or 0�T ,

1 1

Equation of the chord of the hyperbola whose mid

point (x1, y

1) is given

1 1

2 2

3 3

Equation of the chord of the hyperbola 12

2

2

2

��b

y

a

x,

bisected at the given point ),( 11 yx is

12

1

2

1 ��b

yy

a

xx= 1

2

21

2

21 ��

b

y

a

xi.e., 1ST � .

Equation of the chord joining two points on the

hyperbola

The equation of the chord joining the points

)tan,sec( 11 �� baP and )tan,sec( 22 �� baQ is

��

��

��

�

��

��

�

��

�

2cos

2sin

2cos 212121 ��

b

y

a

x.

Rectangular or equilateral hyperbola

(1) Definition : A hyperbola whose asymptotes are at right

angles to each other is called a rectangular hyperbola.

The eccentricity of rectangular hyperbola is always 2 .

The general equation of second degree represents a

rectangular hyperbola if �� 0, abh �2 and coefficient

of 2x + coefficient of 2y = 0.

(2) Parametric co-ordinates of a point on the

hyperbola XY = c2 : If t is non–zero variable, the

coordinates of any point on the rectangular hyperbola

2cxy � can be written as )/,( tcct . The point t )/,( tcct on

the hyperbola 2cxy � is generally referred as the point

‘t’.

For rectangular hyperbola the coordinates of foci are

)0,2( a� and directrices are 2ax �� .

For rectangular hyperbola 2cxy � , the coordinates of

foci are )2,2( cc �� and directrices are 2cyx �� .

(3) Equation of the chord joining points t1 and t

2 : The

equation of the chord joining two points

��

��

��

��

22

11 ,and,

t

cct

t

cct on the hyperbola 2cxy � is

)( 112

12

1

ctxctct

t

c

t

c

t

cy �

�

�� )( 2121 ttcttyx �� .

(4) Equation of tangent in different forms

(i) Point form : The equation of tangent at ),( 11 yx to the

hyperbola 2cxy � is 211 2cyxxy �� or 2

11

��y

y

x

x.

(ii) Parametric form : The equation of the tangent at ��

��

t

cct,

to the hyperbola 2cxy � is cytt

x2�� .On replacing

1x by ct and 1y by t

c on the equation of the tangent at

),( 11 yx

i.e., 211 2cyxxy �� we get cyt

t

x2�� .

Point of intersection of tangents at '' 1t and '' 2t is

��

��

�� 2121

21 2,

2

tt

c

tt

tct.

(5) Equation of the normal in different forms :

(i) Point form : The equation of the normal at ),( 11 yx too

the hyperbola 2cxy � is 21

2111 yxyyxx �� .

(ii) Parametric form : The equation of the normal at ��

��

t

cct,

to the hyperbola 2cxy � is 043 �� cctytxt .

On replacing 1x by ct and 1y by tc / in the equation.

We obtain ,21

2111 yxyyxx ��

043

2

222 �� cctytxt

t

ctc

t

ycxct .

This equation is a fourth degree in t. So, in general four

normals can be drawn from a point to the hyperbola

2cxy � , and point of intersection of normals at 1t and

2t is

��

��

�

��

�

��

)(

)}({,

)(

}1)({

2121

22121

32

31

2121

2221

2121

tttt

ttttttc

tttt

ttttttc.

Mathematics

1. The locus of the intersection point of

ayx �� sincos and byx �� cossin is

(a) Ellipse (b) Hyperbola

(c) Parabola (d) None of these

2. The length of the latus rectum of the parabola whose

focus is (3, 3) and directrix is 0243 �� yx is

(a) 2 (b) 1


3. 05222 �� yxy represents

(a) A circle whose centre is (1, 1)

(b) A parabola whose focus is (1, 2)

(c) A parabola whose directrix is 2

3�x

(d) A parabola whose directrix is 2

1��x

4. The equation of the directrix of the parabola

02442 �� xyy is

(a) 1��x (b) 1�x

(c)2

3��x (d)

2

3�x

5. The curve described parametrically by 12 �� ttx ,

12 �� tty represents

(a) A pair of straight lines (b) An ellipse

(c) A parabola (d) A hyperbola

6. The point of intersection of tangents at the ends of the

latus rectum of the parabola xy 42 � is

(a) (1, 0) (b) (–1, 0)

(c) (0, 1) (d) (0, –1)

7. If the tangents at P and Q on a parabola meet in T, then

SP, ST and SQ are in

(a) A.P. (b) G.P.

(c) H.P. (d) None of these

8. The angle between tangents to the parabola axy 42 �

at the points where it intersects with the line

0�� ayx , is

(a)3

�(b)

4

�

(c)6

�(d)

2

�

9. The tangents drawn from the ends of latus rectum of

xy 122 � meets at

(a) Directrix (b) Vertex

(c) Focus (d) None of these

10. If the tangent and normal at any point P of a parabola

meet the axes in T and G respectively, then

(a) SPSGST �� (b) SPSGST ��

(c) SPSGST �� (d) SPSGST .�

11. The equation of kyxyx �� 3518832 22

represents

(a) No locus if 0�k (b) An ellipse, if 0�k

(c) A point if, 0�k (d) A hyperbola, if 0�k

12. The number of points of intersection of the two

curves xy sin2� and 325 2 �� xxy is

(a) 0 (b) 1

(c) 2 (d) �

13. If the chord joining the points )2,( 121 atat and )2,( 2

22 atat

of the parabola axy 42 � passes through the focus of

the parabola, then

(a) 121 ��tt (b) 121 �tt

(c) 121 �� tt (d) 121 �� tt

14. The locus of the midpoint of the line segment joining

the focus to a moving point on the parabola axy 42 � is

another parabola with the directrix

(a) ax �� (b)2

ax ��

(c) 0�x (d)2

ax �

15. On the parabola 2xy � , the point least distance from

the straight line 42 �� xy is

(a) (1, 1) (b) (1, 0)

(c) (1, –1) (d) (0, 0)

Mathematics

16. The length of the latus-rectum of the parabola whose

focus is ��

��

� �� 2cos

2,2sin

2

22

g

u

g

u and directrix is

g

uy

2

2

� ,

is

(a) �22

cosg

u(b) �2cos

2

g

u

(c) �2cos2 2

2

g

u(d) �2

2

cos2

g

u

17. The line 01 ��x is the directrix of the parabola

082 �� kxy . Then one of the values of k is

(a)8

1(b) 8

(c) 4 (d)4

1

18. The centre of the circle passing through the point (0, 1)

and touching the curve 2xy � at (2, 4) is

(a) ��

��

�

10

27,

5

16(b) �

�

��

�

10

5,

7

16

(c) ��

��

�

10

53,

5

16(d) None of these

19. Consider a circle with its centre lying on the focus of the

parabola pxy 22 � such that it touches the directrix of

the parabola. Then, a point of intersection of the circle

and the parabola is

(a) ��

��

p

p,

2 (b) ��

��

� p

p,

2

(c) ��

��

�p

p,

2 (d) ��

��

�

�p

p,

2

20. Which one of the following curves cuts the parabola

axy 42 � at right angles

(a) 222 ayx �� (b) axey 2/��

(c) axy � (d) ayx 42 �

21. The angle of intersection of the curves �/22 xy � and

xy sin� , is

(a) )/1(cot 1 �� (b) �1cot �

(c) )(cot 1 �� (d) )/1(cot 1 ��

22. The equation of the common tangent to the curves

xy 82 � and 1��xy is

(a) 293 �� xy (b) 12 �� xy

(c) 82 �� xy (d) 2�� xy

23. The equation of the parabola whose focus is the point

(0, 0) and the tangent at the vertex is 01 �� yx is

(a) 0444222 �� yxxyyx

(b) 0444222 �� yxxyyx

(c) 0444222 �� yxxyyx

(d) 0444222 �� yxxyyx

24. If 0�a and the line 0432 �� dcybx passes through

the points of intersection of the parabolas axy 42 � and

ayx 42 � , then

(a) 0)23( 22 �� cbd (b) 0)23( 22 �� cbd

(c) 0)32( 22 �� cbd (d) 0)32( 22 �� cbd

25. The locus of mid point of that chord of parabola which

subtends right angle on the vertex will be

(a) 082 22 �� aaxy (b) )4(2 axay ��

(c) )4(42 axay �� (d) 043 22 �� aaxy

26. The equation of a circle passing through the vertex and

the extremities of the latus rectum of the parabola

xy 82 � is

(a) 01022 �� xyx (b) 01022 �� yyx

(c) 01022 �� xyx (d) 0522 �� xyx

27. The centre of an ellipse is C and PN is any ordinate and A,

A’ are the end points of major axis, then the value of

NAAN

PN

'.

2

is

(a)2

2

a

b(b)

2

2

b

a

(c) 22 ba � (d) 1

28. Let P be a variable point on the ellipse 12

2

2

2

��b

y

a

x

with foci 1F and 2F . If A is the area of the triangle 21FPF ,

then maximum value of A is

(a) ab (b) abe

(c)ab

e(d)

e

ab

Mathematics

29. A man running round a race-course notes that the sum

of the distance of two flag-posts from him is always 10

metres and the distance between the flag-posts is 8

metres. The area of the path he encloses in square metres

is

(a) �15 (b) �12

(c) �18 (d) �8

30. If the angle between the lines joining the end points of

minor axis of an ellipse with its foci is 2/� , then the

eccentricity of the ellipse is

(a) 1/2 (b) 2/1

(c) 2/3 (d) 22/1

31. The eccentricity of an ellipse, with its centre at the origin,

is 2

1. If one of the directrices is 4�x , then the equation

of the ellipse is

(a) 134 22 �� yx (b) 1243 22 �� yx

(c) 1234 22 �� yx (d) 143 22 �� yx

32. The line pyx �� sincos will be a tangent to the

conic 12

2

2

2

��b

y

a

x, if

(a) �� 22222 cossin bap ��

(b) 222 bap ��

(c) �� 22222 cossin abp ��

(d) None of these

33. The angle of intersection of ellipse 12

2

2

2

��b

y

a

x and

circle abyx �� 22 , is

(a) ��

��

��

ab

ba1tan (b) ��

��

��

ab

ba1tan

(c) ��

��

��

ab

ba1tan (d) ��

��

��

ab

ba1tan

34. On the ellipse 194 22 �� yx , the points at which the

tangents are parallel to the line yx 98 � are

(a) ��

��

5

1,

5

2(b) �

�

��

�

5

1,

5

2

(c) ��

��

��

5

1,

5

2(d) �

�

��

�

5

1,

5

2

35. The area of the quadrilateral formed by the tangents at

the end points of latus rectum to the ell ipse

159

22

��yx

, is

(a) 27/4 sq. unit (b) 9 sq. unit

(c) 27/2 sq. unit (d) 27 sq. unit

36. Tangent is drawn to ell ipse 127

22

�� yx

at

)sin,cos33( �� where )2/,0( �� / . Then the value of

� such that sum of intercepts on axes made by this

tangent is minimum, is

(a) 3/� (b) 6/�

(c) 8/� (d) 4/�

37. The locus of the middle point of the intercept of the

tangents drawn from an external point to the ellipse

22 22 �� yx between the co-ordinates axes, is

(a) 12

1122��

yx(b) 1

2

1

4

122��

yx

(c) 14

1

2

122��

yx(d) 1

1

2

122��

yx

38. If the normal at any point P on the ellipse cuts the major

and minor axes in G and g respectively and C be the

centre of the ellipse, then

(a) 2222222 )()()( baCgbCGa ��

(b) 2222222 )()()( baCgbCGa ��

(c) 2222222 )()()( baCgbCGa ��

(d) None of these

39. The locus of the poles of normal chords of an ellipse is

given by

(a)222

2

6

2

6

)( bay

b

x

a�� (b)

222

2

3

2

3

)( bay

b

x

a��

(c)222

2

6

2

6

)( bay

b

x

a�� (d)

222

2

3

2

3

)( bay

b

x

a��

40. If � and � are eccentric angles of the ends of a pair of

conjugate diameters of the ellipse 12

2

2

2

��b

y

a

x, then

�� is equal to

(a)2

�� (b) ��


Mathematics

41. If PQ is a double ordinate of hyperbola 12

2

2

2

��b

y

a

x

such that OPQ is an equilateral triangle, O being the

centre of the hyperbola. Then the eccentricity e of the

hyperbola satisfies

(a) 3/21 �� e (b) 3/2�e

(c) 2/3�e (d) 3/2�e

42. Equation �cos8

3

8

11��

r represents

(a) A rectangular hyperbola

(b) A hyperbola

(c) An ellipse (d) A parabola

43. If the two tangents drawn on hyperbola 12

2

2

2

��b

y

a

x in

such a way that the product of their gradients is 2c , then

they intersects on the curve

(a) )( 22222 axcby �� (b) )( 22222 axcby ��

(c) 222 cbyax �� (d) None of these

44. C the centre of the hyperbola 12

2

2

2

��b

y

a

x. The tangents

at any point P on this hyperbola meets the straight lines

0�� aybx and 0�� aybx in the points Q and R

respectively. Then �CRCQ .

(a) 22 ba � (b) 22 ba �

(c) 22

11

ba� (d) 22

11

ba�

45. If 9�x is the chord of contact of the hyperbola

922 �� yx , then the equation of the corresponding

pair of tangents is

(a) 091889 22 �� xyx

(b) 091889 22 �� xyx

(c) 091889 22 �� xyx

(d) 091889 22 �� xyx

46. Let )tan,sec( �� baP and )tan,sec( �� baQ , where

2

�� , be two points on the hyperbola 1

2

2

2

2

��b

y

a

x.

If (h, k) is the point of intersection of the normals at P and

Q, then k is equal to

(a)a

ba 22 �(b) �

��

��

��

a

ba 22

(c)b

ba 22 �(d) �

��

��

��

b

ba 22

47. The combined equation of the asymptotes of the

hyperbola 054252 22 �� yxyxyx

(a) 0252 22 �� yxyx

(b) 0254252 22 �� yxyxyx

(c) 0254252 22 �� yxyxyx

(d) 0254252 22 �� yxyxyx

48. An ellipse has eccentricity 2

1 and one focus at the point

��

��

1,

2

1P . Its one directrix is the common tangent nearer

to the point P, to the circle 122 �� yx and the hyperbola

122 �� yx . The equation of the ellipse in the standard

form, is

(a) 112/1

)1(

9/1

)3/1( 22

��

�� yx

(b) 112/1

)1(

9/1

)3/1( 22

��

�� yx

(c) 112/1

)1(

9/1

)3/1( 22

��

�� yx

(d) 112/1

)1(

9/1

)3/1( 22

��

�� yx

49. If a circle cuts a rectangular hyperbola 2cxy � in A, B, C,

D and the parameters of these four points be 321 ,, ttt

and 4t respectively. Then

(a) 4321 tttt � (b) 14321 �tttt

(c) 21 tt � (d) 43 tt �

50. The equation of common tangents to the parabola

xy 82 � and hyperbola 33 22 �� yx , is

(a) 012 �� yx (b) 012 �� yx

(c) 012 �� yx (d) 012 �� yx

Mathematics

1. (d) 2. (a) 3. (c) 4. (d) 5. (c) 6. (b) 7. (b) 8. (d) 9. (a) 10. (d)

11. (c) 12. (a) 13. (a) 14. (c) 15. (a) 16. (d) 17. (c) 18. (c) 19. (ab) 20. (b)

21. (b) 22. (d) 23. (c) 24. (d) 25. (a) 26. (c) 27. (a) 28. (b) 29. (a) 30. (b)

31. (b) 32. (c) 33. (d) 34. (bd) 35. (d) 36. (b) 37. (c) 38. (a) 39. (a) 40. (a)

41. (d) 42. (b) 43. (a) 44. (a) 45. (b) 46. (d) 47. (d) 48. (a) 49. (b) 50. (a)

Crash Course Coordinate English

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