Post on 11-May-2017
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
(c) Parabola (d) Ellipse
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
(c) 19117 �� yx (d) None of these
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
(c) 19222 �� yx (d) None of these
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
(c) 0323 ��� yx (d) None of these
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
(c) Parabola (d) Ellipse
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
(c) 0 (d) None of these
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
(c) 10 (d) None of these
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
Equations of tangent in different forms
(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 ��� .
Equations of normal in different forms
(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 �� .
Equations of tangent in different forms
(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 ��� .
Equations of normal in different forms
(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
(c) 4 (d) None of these
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) ��
(c) 0 (d) None of these
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)