Mathematics

EllipseSession - 1

Session Objectives

Session Objectives

1. Introduction

2. Standard form of ellipse

3. Definition of special points or lines

4. Definition in form of focal length

5. Parametric form, eccentric angle

6. Position of point with respect to ellipse

7. Intersection of line and ellipse

8. Condition for tangency

9. Equation of tangent in slope form, point of contact

Ellipse

(i) Fixed point is known as focus (S).

(ii) Fixed line is known as directrix (DD´).

(iii) Fixed ratio is known as eccentricty (e).

0 < e < 1

PS

i.e. ePN

It is the locus of a point P(h, k) inx-y plane which moves such that theratio of its distance from a fixed pointto its distance from a fixed straightline is constant.

N

D

D´

P

S ( Focus) Dir

ectr

ix

Equation of Ellipse in Standard Form

2 2

2 2

2 2

x y+ =1 a > b

a b

y

xO

x´

y´

A´ S´ S(ae, o)

M A Z

N

D

D´

(h, k)P

Q

Equation of Ellipse in Second Form

As we have already discussed

that in the equation

,

then the major and minor axes lie along x-axis and y-axis respectively.

But if , then the major axis of the ellipse lies along y-axis and is of length 2b and minor axis along the x-axis and is of length 2a.

2 2

2 22 2

x y1, if a b

a b

2 2a b

y

xO

x´

y´

A(– a, 0)

A´( a , 0)

S (0, be)

S´ (0, – be)

B (0, b)

B´ (0, – b)

Definition of Special Points/Lines of the Ellipse

y

xO

x´

y´

A´ S´ R A Z

N

(x, y)PQ

Q´

N´

Z´ S

L

L´

x = – —ae

x = —ae

B´

B Ellipse

2 2

2 2

x y1,

a ba b

Ellipse

2 2

2 2

x y1,

a ba b

(I) Coordinates of the centre (0, 0) (0, 0)

(II) Coordinates of the vertices a, 0 0, b

(III) Coordinates of foci ae, 0 0, be

(IV) Length of major axis 2a 2b

y

xO

x´

y´

A´ S´ R A Z

N

(x, y)PQ

Q´

N´

Z´ S

L

L´

x = – —ae

x = —ae

B´

B Ellipse

2 2

2 2

x y1,

a ba b

Ellipse

2 2

2 2

x y1,

a ba b

(V) Equation of the directrices a

xe

b

ye

(VII) Equation of minor axis x = 0 y = 0

(VI) Equation of major axis y = 0 x = 0

Definition of Special Points/Lines of the Ellipse

y

xO

x´

y´

A´ S´ R A Z

N

(x, y)PQ

Q´

N´

Z´ S

L

L´

x = – —ae

x = —ae

B´

B Ellipse

2 2

2 2

x y1,

a ba b

Ellipse

2 2

2 2

x y1,

a ba b

(VI) Eccentricity

(VII) Length of the latus rectum

2 2 2b a 1 e 2 2 2a b 1 e

22b

a

22a

b

Definition of Special Points/Lines of the Ellipse

Focal Distance of a Point on the Ellipse

Let P(x, y) be any point on the ellipse

Then SP = ePN

a= e(RZ) = e(OZ OR) e x

e

y

xO

x´

y´

A´ S´ R A Z

N

(x, y)PQ

Q´

N´

Z´ S

L

L´

x = – —ae

x = —ae

B´

B

Focal Distance of a Point on the Ellipse

SP = a – ex ... (i)and S´P = ePN´

= e(RZ´) = e(OR + OZ´)

ae x

e

S´P a ex ...(ii)

= Major axis SP S´P 2a

“An ellipse is the locus of a point which moves in such a way that the sum of its distances from two fixed points (foci) is always constant.”

On the basis of above property, the definition of ellipse can be given as follows.

General Equation of Ellipse

If the centre of the ellipse is at point(h, k) and the axes of ellipse is parallelto the coordinate axes, then its

equation is .

2 2

2 2

x h y k1

a b

Parametric Form of Ellipse

2 2

2 2

x y+ =1, a > b

a b

Auxiliary circle

y

xO

x´

y´

x

a

y

b

2

2

2

2— + — = 1

(a > b )2 2x + y = 2 2 a2

(Auxiliary circle)

The circle described on themajor axis of an ellipse asdiameter is called an auxiliarycircle of the ellipse.

If is an

ellipse, then its auxiliary circle is x2 + y2 = a2.

2 2

2 22 2

x y+ = 1 a > b

a b

Eccentric Angle of Point y

xO

x´

y´

A´ A

Q

P

N

Q lies on the circle

, 2 2 2x y a

coordinate of

Q acos , bsin

Parametric Coordinates of a Point on an Ellipse

Let coordinates of P be . p px , y

px a cos

py ?

p px , y lies on ellipse 2 2

2 2

x y1

a b

2 2p p

2 2

x y1

a b

22p 22 2

y acos1 sin

b a py bsin

P acos , bsin

Equation of Chord

2 2

2 2

Let the equation of ellipse

x ybe 1, a b

a b

A a cos , bsin

B a cos , bsin

y

xO

x´

y´

B(a cos , b sin )

B (a cos , b sin )

Equation of Chord

B A

B AB A

y yEquation of line AB is y y x x

x x

bsin bsin

or y bsin x asinacos acos

2sin cosb 2 2or y sin . x asina 2sin sin

2 2

sin cos2 2

or y bsin x acosb a

Equation of Chord

yor sin sin sin

b 2 2

xcos cos cos

a 2 2

x ycos sin cos cos sin sin

a 2 b 2 2 2

x ycos sin cos

a 2 b 2 2

Position of a point w.r.t. ellipse

2 2x y

Consider: 1 025 9

2 2x y

Let E x, y 1 be an expression.25 9

E (0, 0) = –1 = –ve

E (0, 1) = 1

1 ve9

i.e. (i) If point (x1, y1) lies inside the ellipse, then E(x1, y1) < 0.

(ii) If point (x1, y1) lies on the ellipse, then E(x1, y1) = 0.

(iii) If point (x1, y1) lies outside the ellipse, then E(x1, y1) > 0.

Intersection of Line and Ellipse

AB

x

y

O

m x – y + c = 0

x2

a2—— + —— = 1(a2 2> b )y2

b2

Let line mx – y + c = 0 and ellipse intersect

at the distinct points A .

2 2

2 2

x y1

a b A A B Bx , y and B x , y

Intersection of Line and Ellipse

A AThen mx y c 0

2 2A A2 2

x y1 0

a b

B Bmx y c 0

2 2B B2 2

x y1 0

a b

2 2

2 2

x yi.e. if we substitute 'y' from y = mx + c into 1,

a b

we get a quadratic in 'x', whose roots are .A Bx and x

Intersection of Line and Ellipse

If discriminant of that quadratic > 0,then the line intersect the ellipseat two distinct points.

If discriminant of that quadratic = 0,the line touches the ellipse.

If discriminant of that quadratic < 0,the line does not cut the ellipse.

Intersection of Line and Ellipse

Now, let us consider the case whenD = 0.

x

y

O

m x – y + c = 0

x2

a2—— + —— = 1y2

b2

22

2 2

mx cx1

a b

22 2 2 2 2b x a mx c a b

2 2 2 2 2 2 2 2 2or x a m b 2a mcx a c a b 0

2D 0 B 4AC 0

Intersection of Line and Ellipse

4 2 2 2 2 2 2 2 24a m c 4 a m b a c b 0

2 2 2 2c a m b

2 2 2c a m b

2 2 2y mx a m b is always tangent to the ellipse

for all values of m R.

Point of contact of a tangent with2 2

2 2

x y+ = 1

a b

Let us consider again the equation

. 2 2 2 2 2 2 2 2x a m b 2a mcx a c b 0

Let the point of contact be , i.e. c cx , y

2

A B c 2 2 2

2a mcx x 2x

a m b

or

2

c 2

a mcx

c

2a m

c

Point of contact of a tangent with2 2

2 2

x y+ = 1

a bAlso y = mx + c is tangent line, passing

through point of contact . c cx , y

2

c ca m

y mx c m cc

2 2 2 2a m c b

c c

2 2a m b Point of contact is , .

c c

2 2 2

2 2

1

i.e. the tangent y mx a m b touches

a m bthe ellipse at point T , ,

c cwhere 2 2 2c a m b 0

Point of contact of a tangent with2 2

2 2

x y+ = 1

a bso that (if m is positive)T1

x ve

(if m is positive)T1y ve

x

y

O

T1

T2

and the tangent

touches the ellipse at point

where

so that (if ‘m’ is positive),

(if ‘m’ is positive).Also we note that these two tangentsare parallel.

2 2 2y mx a m b

2 2

2 2

x y1

a b

2 2

2a m b

T , ,c c

2 2 2c a m b 0

T2x ve

T2y ve

Class Exercise - 1

Find the centre, vertices, lengths of axes, eccentricity, coordinates of foci, equations of directrices, and length of latus-rectum of the ellipse

2 24x 16y 24x 32y 12 0.

Solution

We have 2 24x 16y 24x 32y 12 0

2 24 x 6x 16 y 2y 12

2 24 x 6x 9 16 y 2y 1 12 36 16

2 2x 3 y 1

1 ....(i)16 4

Shifting the origin at (3, 1) without rotating the coordinate axes, i.e.

put X = x – 3 and Y = y – 1

Solution contd..

Equation (i) reduces to

2 2X Y

1 ....(ii)16 4

2 2Here a 16, b 4

Clearly, a > b. Therefore, the given equation represents an ellipse whose major and minor axes are along X-axis and Y-axis respectively.

Centre: The coordinates of the centre with respect to new axes are X = 0 and Y = 0.

Coordinates of the centre with respect to old axes are x – 3 = 0 and y – 1 = 0, i.e. (3, 1).

Solution contd..

Vertices: The coordinates of vertices with respect to the new axes are

X a, Y 0 i.e. X 4, Y 0

The vertices with respect to the old axesare given by

x 3 4 and y 1 = 0, i.e. (7, 1) and ( 1, 1).

Lengths of axes: Here a = 4, b = 2

Length of major axis = 2a = 8

Length of minor axis = 2b = 4

Solution contd..

Eccentricity: The eccentricity e is given by

2

2

be 1

a

4 3

116 2

Coordinates of foci: The coordinates of foci withrespect to new axes are X ae, Y 0 i.e. X 2 3, Y 0

Coordinates of foci with respect to old axes are

x 3 2 3, y 1 0 i.e. 3 2 3, 1

Solution contd..

Equation of directrices: The equationof directrices with respect to new axes

are ,a

Xe

4 2i.e. X

3

Equation of directrices with respect to old

axes are 8 8

x 3 i.e. x 3 .3 3

22b 2 4

Length of latus rectum 2a 4

Class Exercise - 2

Find the equation of the ellipse whose axes are parallel to the coordinate axes respectively having its centre at the point (2, –3), one focus at (3, –3) and one vertex at (4, –3).

Solution

Let 2a and 2b be the major and minor axes of the ellipse. Then its equation is

2 2

2 2

x 2 y 31

a b

As we know that distance between centre andvertex is the semi-major axes,

2 24 2 3 3 a 2.

Again, since the distance between the focus andcentre is equal to ae,

2 23 2 3 3 ae

Solution contd..

1

1 2e e2

Again 2 2 2b a 1 e

14 1 3

4

Equation of ellipse is

2 2x 2 y 3

14 3

2 2i.e. 3x 4y 12x 24y 36 0

Class Exercise - 3

An ellipse has OB as a

semi minor axis. F and F´ are its foci and is a right angle. Find the eccentricity of ellipse.

2 2

2 2

x y1

a b

FBF´

Solution

xO

y

FF´

B

x´

y´The equation of the ellipse is

2 2

2 2

x y1

a b

Coordinates of F and F´ are (ae, 0) and (–ae, 0)respectively. Coordinates of B are (0, b).

Solution contd..

Slope of BF =

b 0 b0 ae ae

and slope of BF´ =

b 0 b0 ae ae

F´BF is right angle,

2 2 2b b1 b a e

ae ae

2 2 2 2 2a 1 e a e 2e 1

1

e2

Class Exercise - 4

Let P be a variable point on the ellipse with foci at S and S´. If A

be the area of PSS´, find the maximum value of A.

2 2x y

125 16

Solution

Here equation of ellipse is 2 2x y

125 16

a 5, b 4

Coordinates of P can be taken as 5 cos , 4sin

xO

y

SS´

P (5cos , 4sin )

x´

y´

M

2 2 2 2b a 1 e 16 25 1 e

3

e5

Solution contd..

Coordinates of S 3, 0 and S´ 3, 0

1

Area of PSS´ SS´ PM2

1

6 4sin 12 sin2

Maximum area = 12 sq. unit as maximum value of sin 1

Class Exercise - 5

Find the equation of tangents to the

ellipse which cut off

equal intercepts on the axes.

2 2

2 2

x y1

a b

Solution

In case of tangent makes equal interceptmakes equal intercepts on the axes, then

it is inclined at an angle of to X-axisand hence its slope is

45

m tan 45 1

Equation of tangent is 2 2y x a b

Class Exercise - 6

Find the equation of tangent to the ellipse which are (i) parallel, (ii) perpendicular to the liney + 2x = 4.

2 23x 4y 12

Solution

Equation of ellipse can be written

2 2x y

14 3

2 2a 4, b 3

Slope of the line y = –2x + 5 is –2.

Any tangent to the ellipse is

2 2 2y mx a m b

If the tangent is parallel to the given line, slope of tangent is –2.

Solution contd..

Equation of tangent is y 2x 4 4 3

i.e. y 2x 19

If the tangent is perpendicular to the given line,

slope of tangent is .12

Equation of tangent is 1

y x 192

Class Exercise - 7

Prove that eccentric angles of the extremities of latus recta of the

ellipse are given by 2 2

2 2

x y1

a b

1 btan .

ae

Solution

Let be the eccentric angle of an end of a latus rectum. Then the coordinates of

the end of latus rectum is .

As we know that coordinates of latus

rectum is ,

acos , bsin

2bae,

a

a cos ae cos e and 2b b

bsin sina a

b

tanae

1 btan

ae

Class Exercise - 8

A circle of radius r is concentric with the

ellipse Prove that the common

tangent is inclined to the major axis at

an angle .

2 2

2 2

x y.

a b

2 21

2 2

r btan

a r

Solution

Equation to the circle of radius r and concentric with ellipse whose centre is (0, 0) is

2 2 2x y r ....(i)

Any tangent to the ellipse is

2 2 2y mx a m b ....(ii)

If it is a tangent to circle, then perpendicular fromcentre (0, 0) is equal to r.

2 2 2

2

a m br

1 m

Solution contd..

2 2 2 2 2a m b r 1 m

2 2 2 2 2or m a r r b

2 2 2 2 2 22

2 2 2 2 2 2

r b r b r bm m tan

a r a r a r

2 21

2 2

r btan

a r

Class Exercise - 9

If the line lx + my + n = 0 will cut the

ellipse in points whose

eccentric angles differ by then prove

that

,

2

2 2

2 2

x y1

a b

2 2 2 2 2a l b m 2n .

Solution

Let the line lx + my + n = 0 cuts theellipse at

P acos , bsin and Q acos , bsin2 2

These two points lie on the line lx + my + n = 0

al cos bm sin n

and al sin bm cos n

Squaring and adding,

2 2 2 2al cos bmcos alsin bmcos n n

2 2 2 2 2a l b m 2n (Pr oved)

Class Exercise - 10

Find the locus of the foot of the perpendicular drawn from centre upon

any tangent to the ellipse 2 2

2 2

x y1.

a b

Solution

Any tangent to the given ellipse is

2 2 2y mx a m b ....(i)

Equation of any line perpendicular to (i), passingthrough the origin is

1

y x ...(ii)m

xC

y

P

x´

y´

Solution contd..

In order to find the locus of P, the point of intersection of (i) and (ii), we have to eliminate m.

2 2

22

x a xy .x b

y y

2 2 2 22 a x b yxor y

y y

2 2 2 2 2 2y x a x b y 22 2 2 2 2 2x y a x b y

This is the required locus.

Thank you