Conic Section.pdf

5/27/2018 Conic Section.pdf

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SYLLABUS JEE MAINS

CONIC SECTION

Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) in standard

forms, condition for y = mx + c to be a tangent and point (s) of tangency.

QUICK REVISION

1. DEFINITION

A parabola is the locus of a point which moves in a plane such that its distance from a

fixed point (called the focus) is equal to its distance from a fixed straight line (called the

directrix).

2. Terms related to parabola

Axis:A straight line passes through the focus and perpendicular to the directrix is called

the axis of parabola.

Vertex :The point of intersection of a parabola and its axis is called the vertex of theparabola.

The vertex is the middle point of the focus and the point of intersection of axis and

directrix.

S(a, 0)

y = 4ax2

L

L'

A

Y

Q

NP(x, y)

X


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Eccentricity :If P be a point on the parabola and PN and PS are the distance from the

directrix and focus S respectively then the ratio PS/PN is called the eccentricity of the

parabola which is denoted by e.

By the definition for the parabola e = 1.

If e > 1 hyperbola, e = 0 circle, e < 1 ellipse

Latus Rectum

Let the given parabola be y2= 4ax. In the figure LSL' (a line through focus to axis) is

the latus rectum.

Also by definition, LSL' = 2 (4a.a)= 4a

Double ordinate

Any chord of the parabola y2= 4ax which is to its axis is called the double ordinate)

through the focus S.

Focal Chord

Any chord to the parabola which passes through the focus is called a focal chord of the

parabola.

3. Some Standard forms of parabola

(1) Parabola opening to left (2) Parabola opening upwards (3) Parabola opening down

wards

(i.e. 2y 4ax ) (a> 0) (i.e. 2x 4ay) ; (a>0) (i.e. 2x 4ay); (a > 0)

yN

Directrix

Vertex

x

+

a=0

Q A

y'

L'

L' (a, -2a)

Latus Rectum

axisx

Focal chord

Double ordinatex = a

L(a, 2a)L

PFocaldistance

Focus

S (a, 0)


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Important terms 42y ax 42y ax 42x ay 42x a

Coordinates of vertex (0, 0) (0, 0) (0, 0) (0, 0)

Coordinates of focus (a, 0) (a, 0) (0, a) (0,a)

Equation of the directrix x a x a y a y= a

Equation of the axis y 0 y 0 x 0 x 0

Length of the latusrectum 4a 4a 4a 4a

Focal distance of a point P(x,y) x a a x y a a y

4. REDUCTION OF STANDARD EQUATION

If the equation of a parabola contains second degree term either in y or in x(but not in

both) then it can be reduced into standard form. For this we change the given equation

into the following forms-

(yk)2= 4a (xh) or (xp)2= 4b (yq)

Then we compare from the following table for the results related to parabola.

Equation of

Parabola

Vertex Axis Focus Directrix Equation

of L.R.

Length

L.R.

2(y K) 4a(x h) (h,k) y k (h a, k ) x a h 0

x a h 4a

2(x p) 4b(y q) (p,q) x p (p,b q ) y b q 0

y b q 4b


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5. PARAMETRIC EQUATIONS OF A PARABOLA

The parametric equation of the parabola y2= 4ax are x = at2, y = 2at, where t is the

parameter.

6. CONDITION FOR TANGENCY AND POINT OF CONTACT

The line y = mx + c touches the parabola y2 = 4ax if c =a

mand the coordinates of the

point of contact are .

Note

The line y = mx + c touches parabola x

2

= 4ay if c =

am

2

The line x cos + y sin = p touches the parabola y2= 4ax if asin2 + p cos =

0.

7. EQUATION OF TANGENT IN DIFFERENT FORMS

7.1 Point Form

The equation of the tangent to the parabola y2= 4ax at the point (x1, y1) is yy1= 2a (x +

x1)

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

Equation of parabolas Tangent at (x1, y1)

2y 4ax 1 1yy 2a(x x )

2x 4ay 1 1xx 2a(y y )

2x 4ay 1 1xx 2a(y y )

m

a2,

m

a2


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7.2 Parametric Form

The equation of the tangent to the parabola y2= 4ax at the point (at2, 2at) is ty = x +

at2.

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

Equations of

parabolas

Parametric co-ordinates

't'

Tangent at 't'

2y 4ax 2( at ,2at) 2ty x at

2x 4ay 2(2at,at ) 2tx y at

2x 4ay 2(2at, at ) 2tx y at

7.3 Slope Form

The equation of tangent to the parabola y2= 4ax in terms of slope 'm' is y = mx +a

m.

The coordinate of the point of contact are 2a 2a

, mm

Equation of

parabolas

Point of contact in

terms of slope

(m)

Equation of tangent in

terms of slope

(m)

Condition of

Tangency

2y 4ax 2

a 2a,

mm

ay mx

m

ac

m

2y 4ax 2

a 2a,

mm

ay mxm

acm

2x 4ay 2(2am,am ) 2y mx am 2c am

2x 4ay 2( 2am, am ) 2y mx am 2c am


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P(x, y)

Directrix

N

S(focus)

Q

ELLIPSE

1. DEFINITION

An ellipse is the locus of a point which moves in a plane so that the ratio of its distance

from a fixed point (called focus) and a fixed line (called directrix)

is a constant which is less than one. This ratio is called

eccentricity and is denoted by e. For an ellipse, e < 1.

Let S be the focus, QN be the directrix and P be any point on

the ellipse. Then, by definition,PS

PN= e or

PS = e PN, e < 1, where PN is the length of the perpendicular from P on the directrix

QN.

An Alternate DefinitionAn ellipse is the locus of a point that moves in such a way that

the sum of its distacnes from two fixed points (called foci) is constant.

2. EQUATION OF AN ELLIPSE IN STANDARD FORM

The Standard form of the equation of an ellipse is where a and b are

constants.

Symmetry

(a) On replacing y by y, the above equation remains unchanged. So, the curve is

symmetrical about x-axis.

(b) On replacing x by x, the above equation remains unchanged. So, the curve is

symmetrical about y-axis

)ba(1

b

y

a

x2

2

2

2


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3. TERMS RELATED TO AN ELLIPSE

A sketch of the locus of a moving point satisfying the equation , has

been shown in the figure given above.

Foci

If S and S' are the two foci of the ellipse and their coordinates are (ae, 0) and (ae, 0)

respectively, then distance between foci is given by

SS' = 2ae.

Directrices

If ZM and Z' M' are the two directrices of the ellipse and their equations are x =a

eand x

=a

erespectively, then the distance between directrices is given by ZZ' =

2a

e.

Axes The lines AA' and BB' are called the major axis and minor axis respectively of the ellipse.

The length of major axis = AA' = 2a

The length of minor axis = BB' = 2b

B(0,

b)

P(x, y) M

x=a/e

X

ZA(a,0)

Directrix

B(0,-b)

C N

Minor

Axis

Major Axis

N

S'

N'

Z'X'

Directrix

x=-a/e

M'

Y

A'(-a,0)

L

L'

S

P'

)ba(1b

y

a

x2

2

2

2


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CentreThe point of intersection C of the axes of the ellipse is called the centre of the ellipse. All

chords, passing through C are bisected at C.

Vertices The end points A and A' of the major axis are known as the vertices of the ellipse

A (a, 0) and A' (a, 0)

Focal chord A chord of the ellipse passing through its focus is called a focal chord.

Ordinate and Double Ordinate

Let P be a point on the ellipse. From P, draw PN AA' (major axis of the ellipse) and

produce PN to meet the ellipse at P'. Then PN is called an ordinate and PNP' is called

the double ordinate of the point P.

Latus Rectum

If LL' and NN' are the latus rectum of the ellipse, then these lines are to the major axis

AA' passing through the foci S and S' respectively.

Length of latus rectum = LL' =22b

a= NN.

By definition SP = ePM = e = aex and

Thus implies that distances of any point P(x, y) lying on the ellipse from foci are : (a ex)

and (a + ex). In other words

SP + S'P = 2a

i.e., sum of distances of any point P(x, y) lying on the ellipse from foci is constant.

Eccentricity

Since, SP = e.PM, therefore

,a

b,aeL

2

a

b,ae'L

2

a

b,aeN

2

a

b,ae'N

2

xe

a.exax

e

aeP'S


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SP2= e2PM2 or (xae)2+ (y0)2= e2

(xae)2+ y2= (aex)2 x2+ a2e22aex + y2= a22aex +

e2x2

x2(1e2) + y2= a2(1e2)

On comparing with we get

b2= a2(1e2) or e =

Ellipse 2 2

2 2 1

x y

a b

For a> b For b> a

Centre (0, 0) (0, 0)

Vertices ( a,0) (0, b)

Length of major axis 2a 2b

Length of minor axis 2b 2a

Foci ( ae,0) (0, be)

Equation of directrices x a /e y b/ e

Relation in a, band e 2 2 2b a (1 e ) 2 2 2a b (1 e )

Length of latus rectum 22b

a

22a

b

Ends of latus-rectum 2bae,

a

2a

, be

b

Parametric equations (a cos ,bsin ) (a cos ,bsin )(0 2 )

Focal radii 1SP a ex and 1S'P a ex 1SP b ey and 1S'P b ey

Sum of focal radii

SP S 'P

2a 2b

2

xe

a

.1)e1(a

y

a

x22

2

2

2

2 2

2 2

x y1,

a b

2

2

a

b1


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Distance between foci 2ae 2be

Distance between

directrices

2a/e 2b/e

Tangents at the vertices x=a, x = a y= b, y=b

4. POSITION OF A POINT WITH RESPECT TO AN ELLIPSE

The point P(x1, y1) lies outside, on or inside the ellipse according as

or < 0.

5. CONDITION OF TANGENCY AND POINT OF CONTACT

The condition for the line y = mx + c to be a tangent to the ellipse is that c2=

a2m2+ b2and the coordinates of the points of contact are

Note :

x cos a + y sin a = p is a tangent if p2= a2cos2 + b2sin2 .

lx + my + n = 0 is a tangent if n2= a2l2+ b2m2.

Point form: The equation of the tangent to the ellipse2 2

2 2

x y1

a bat the point 1 1( x , y ) is

1 1

2 2

xx yy1

a b

Slope form: If the line y mx c touches the ellipse2

2 2

x y1

a b, then 2 2 2 2c a m b .

Hence, the straight line 2 2 2y mx a m b always represents the tangents to the

ellipse.

1b

y

a

x2

2

2

2

0,01b

y

a

x2

21

2

21

1b

y

a

x2

2

2

2

222

2

222

2

bma

b,

bma

ma


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Points of contact: Line 2 2 2y mx a m b touches the ellipse2

2 2

x y1

a bat

2 2

2 2 2 2 2 2

a m b,

a m b a m b

Parametric form: The equation of tangent at any point (a cos ,bsin ) is

x ycos sin 1

a b

6. Some Standard Result

The straight line lx my n 0 touches the ellipse2 2

2 2

x y1

a b, if 2 2 2 2 2a l b m m .

The line x cos y sin p touches the ellipse2 2

2 2

x y1

a b, if 2 2 2 2 2a cos b sin p

and that point of contact is2 2a cos b sin

,p p

.

Two tangents can be drawn from a point to an ellipse. The two tangents are real and

distinct or coincident or imaginary according as the given point lies outside, on or inside

the ellipse.

The tangents at the extremities of latus-rectum of an ellipse intersect.

Hyperbola

1. DEFINITION

A hyperbola is the locus of a point which moves in a plane so that the ratio of its

distances from a fixed point (called focus) and a fixed line (called directrix) is a constant

which is greater than one. This ratio is called eccentricity and is denoted by e. For a

hyperbola e > 1.

Let S be the focus, QN be the directrix and P be any point on the hyperbola. Then, by

definition

or PS = e PN, e > 1,

where PN is the length of the perpendicular from P on the directrix QN.

PN

PS


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An Alternate Definition

A hyperbola is the locus of a point which moves in such a way that the difference of its

distances from two fixed points (called foci) is constant.

2. EQUATION OF HYPERBOLA IN STANDARD FORM

The general form of standard hyperbola is

where a and b are constants.

Symmetry

Since only even powers of x and y occur in the above equation, so the curve is

symmetrical about both the axes.

3. TERMS RELATED TO A HYPERBOLA

A sketch of the locus of a moving point satisfying the equation , has been

shown in the figure given above.

Foci If S and S' are the two foci of the hyperbola and their coordinatesd are (ae, 0) and (ae,

0) respectively, then distance between foci is given by SS' = 2ae.

1b

y

a

x2

2

2

2

M'M

B

Axis

Directrix

Conju

gateC

x = a/e

Directrix

A(a,0)

Z

LP(x, y)

Rectum

S(ae, 0)

Latus

x=-a/e

A'

(-a,

0)

S'(-ae, 0)

X'

N

N'

Z'

Y'

B'

L'

X

1b

y

a

x2

2

2

2


13/17

Directries ZM and Z' M' are the two directrices of the hyperbola and their equations are x =

and

x = respectively, then the distance directrices is given by zz' = .

Axes The lines AA' and BB' are called the transverse axis and conjugate axis respectively of

the hyperbola.

The length of transverse axis = AA' = 2a

The length of conjugate axis = BB' = 2b

Centre The point of intersection C of the axes of hyperbola is called the centre of the

hyperbola. All chords, passing through C, are bisected at C.

Vertices The points A (a, 0) and A' (a, 0) where the curve meets the line joining the foci

S and S', are called the vertices of the hyperbola.

Focal Chord A chord of the hyperbola passing through its focus is called a focal chord.

Focal Distances of a Point

The difference of the focal distances of any point on the hyperbola is constant and equal

to the length of the transverse axis of the hyperbola. If P is any point on the hyperbola,

then

S'PSP = 2a = Transverse axis.

e

a

e

a

e

a2


14/17

Latus Rectum If LL' and NN' are the latus rectum of the hyperbola then these lines are

perpendicular to the transverse axis AA', passing through the foci S and S' respectively.

, , , .

Length of latus rectum = LL' = = NN'.

Eccentricity of the HyperbolaWe know that

SP = e PM or SP2= e2PM2or (xae)2+ (y0)2= e2

(xae)2+ y2= (exa)2 x2+ a2e22aex + y2= e2x22aex + a2

x2(e21)y2= a2(e21) .

On comparing with , we get b2= a2(e21) or e =

5. CONJUGATE HYPERBOLA

The hyperbola whose transverse and conjugate axes are respectively

the conjugate and transverse axes of a given hyperbola is called the

conjugate hyperbola of the given hyperbola.

The conjugate hyperbola of the hyperbola.

is

a

b,aeL

2

a

b,ae'L

2

a

b,aeN

2

a

b,ae'N

2

a

b2 2

2

e

ax'N

1)1e(a

y

a

x22

2

2

2

1b

y

a

x2

2

2

2

2

2

a

b1

1b

y

a

x.,e.i1

b

y

a

x2

2

2

2

2

2

2

2

1

b

y

a

x2

2

2

2

Y

S(0, be)

B(0, b) y

Z

C

B(0, -b) y

S'(0,-

be)

X'

Y'


15/17

Hyperbola

Fundamentals

2 2

2 2 1

x y

a b

2 2

2 2 1

x y

a bor

2 2

2 12

x y

a b

Centre (0, 0) (0, 0)

Length of transverse

axis

2a 2b

Length of conjugate

axis

2b 2a

Foci ( ae,0) (0, be)

Equation of directrices x a /e y b/e

Eccentricity 2 2

2

a be

a

2 2

2

a be

b

Length of latus rectum 22b

a

22a

b

Parametric co-

ordinates

(asec ,btan ) , 0 2 (b sec ,a tan ),0 2

Focal radii 1SP ex a &

1S P ex a

1SP ey b & 1S P ey b

Difference of focal radii

(S P SP )

2a 2b

Tangents at the

vertices

x a, x a y b,y b

Equation of the

transverse axis

y 0 x 0


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Equation of the conjugate

axis

x 0 y 0

6. POSITION OF A POINT WITH RESPECT TO A HYPERBOLA

The point P(x1, y1) lies outside, on or inside the hyperbola according as

= 0 or < 0.

7. CONDITION FOR TRANGENCY AND POINTS OF CONTACT

The condition for the line y = mx + c to be a tangent to the hyperbola is that c2

= a2m2b2and the coordinates of the points of contact are

8. EQUATION OF TANGENT IN DIFFERENT FORMS

Point Form

The equation of the tangent to the hyperbola at the point (x1, y1) is

.

Note :The equation of tangent at (x1, y1) can also be obtained by replacing x2by xx1,

y2by yy1, x by ,y by and xy by . This method is used only when the

equation of hyperbola is a polynomial of second degree in x and y.

Parametric Form The eqn of the tangent to the hyperbola at the point (a

sec, b tan) is

1b

y

a

x2

2

2

2

.01b

y

a

x2

21

2

21

1

b

y

a

x2

2

2

2

222

2

222

2

bma

b,

bma

ma

1b

y

a

x2

2

2

2

1b

yy

a

xx2

1

2

1

2

xx 1

2

yy 1 1 1xy x y

2

1b

y

a

x2

2

2

2

1tanb

ysec

a

x


17/17

Slope FormThe equation of tangent to the hyperbola in terms of slope 'm' is

y = mx

1b

y

a

x2

2

2

2

222 bma

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