Ellipse Contents

1. Definition
2. Position of a point w.r to ellipse.
3. Equation of tangents, points of content, Chords.
4. Equation of normal different form.
5. Pair of tangents, Chord of content etc.
6. Pole & Polar.
7. Some standerd properties of an ellipse
 Ellipse

Definetion;- Ellipse is the locus of a point which moves buen that its distance from a
fixed point called focus is ina contant ratio to its distance from a dixed line clled
directions. The ration is called eccentricity and is denoted by e.
For ellipse eccentricity e  1

SP/PM = e
(h-)2 + (k-β)2 = e |sh + bk +c/ a2 +b2|

Ean of allipse
(x-)2 + (y-β)2 = e2 (ax +by +c)2
A 2 + b2

Ex. Find the equation of on allipse whose focus is (-1,1) eccentricity ½ and the direction
is x-y+3 = 0
Sol. SP = e .pm
(h+1)2 + (k-1)2 = ½ h-k+3
2
(x+1) +(y-1) = 1/p (x – y +3)2
2 2

7x2 + 7y2 + 2xy +10x-10y + 7 = 0

Standard equation of ellipse;-

x2 + y2 =1 (ab) b2 = a2(1-e2)
a2 b2

(i) Centre:- All the chord passing through e is bisected at c

C= (0,0)
(ii) Foci:- S and S’ are two foci of the alipse and there co-ordinates are (ae,o) and (-ae, o)
respectively
(iii) Directrices;- x = a/e and x= -a/e ax
Called directrics
(iv) Double ordinates;- PP’ is called a double ordinates.

(v) Latur rectum: LL’ is called latur rectum

Length of lathurectum = 2b2

a

(vi) Axis: The line AA’ and BB’ are called the major and minor Axis
Ean of major Axis y =0
“ “ minor “ x=0

(vii) focal chord:- A chord of ellipse passing through its focus is called a focal chord.

(viii) Vertices :- A and A’ 1 are called vertices A=(a,0 , A’ = (-a, 0)

(ix) Parametric Equation of ellipse;-

X 2 + y2 = 1
A2 b2

Equation of Auxilary civilr

X2 + y2 = a2
R = (a cosθ, asinθ)
P = (a cosθ, y)
X 2 + y2 = 1
A2 b2
A2 Cos2θ + y2 = 1
A2 b2
Y2 = 1- Cos2θ = sin2θ
B2
Y2= b2 sin2θ
Y = + bsinθ (a cosθ, bsinθ)

Focal distance of a point:-

The sum of focal distance of any point on the ellipse is equal to the major axix.
x2 +y2 = 1
a2 b2

SP + S’P = (a-ex1) + (a+ex1)

SP + S’P = 2a = AA’ = Major axis
The Shope of the ellipse x2 / a2 + y2 / b2 = 1 , ba
Direction : y = b/e , y= -b/e
Foci (u, + be)

X2 + y2 =1, ab X2 +y2 = 1, ba

A2 b2 A2 b2
1. Centre :- (0,0) (0,0)
2. Vertices (+a, o) (o, +b)
3. Length of major axis 2a 2b
4. Length of Minor Axis foci 2b 2a
5. Equation of directrix (+ ae, 0) (0, +b/e)
X = +a/e Y = + b/e
6. eccentricity B2 = a2(1-e2) A2 = b2(1-e2)
7.length of lathrectum 2b2/a 2a2/b
8. Ends of latusrectum (ae, b2/a) (a2/b , be)
9. Parametric Co-ordinater (acosθ, bsinθ) (a cosθ , bsinθ)
10. focal ratio SP = a-ex1 & S’P = a + ex1 SP = b-ey1 & S’P = b+ey1
11. Sum of focal ratio SP + 2a 2b
SP’
12. Distance between foci 2ae 2be
13. Dis. Between directrics 2a/e 2b/e
14. Tangents of the vertices X = a, x = -a Y = b, y = -b

Ex. Find the length of major and minor axix the co-ordinates of foci and verticex and the
eccentrecites of the ellipse 3x2 + 2y2 = 6, Also find the equation of the direction.
Ans. 3x2 + 2y2 = 6
X2/2 + y2/3 = 1 ba
Length of major Axix = 23
“ “ minimum “ = 22
If 2 is eccentricity a2 = b2 (1-e2)
2 = 3(1-e2)
E2 = 1/3 , e = 1/3

Vertica = (u, +b) = (0 , +3)

For (u , +be) = (0, +1)
Equation of direction y = +3/e
Y = +3/ 1/3
Y = +3
Example :- Find the equation of the ellipse unferes to its centre when minor axis in equal
to the distance between the foci and whose latur rectum is 10
Sol. Led equation of ellipse x2 + y2 = 1
a2 b2
foci (ae , 0)
length of minor Axis = 2b
length of latur recture = 2b2/a

Given, length of minor Axis = distance between foci

Lb = 2ae
b/a = e  b = ae
& 2b2 = 10  b2 = 5a
a
B = a2(1-e2)
2

A2e2 = a2(1-e2)
e2 = 1-e2  2e2 =1

e=1
2

b=a
2
B = a2
2

2
54 = a2
2
A =10
B2 = 10* 1 = 52
 2
Equ. X + y2 = 1
2

100 50

Ex. The extremities of a line segment of length l more in two tired perpendicular straight
lines. Find the locus of that point which divides this line segment in ratio 1:2

Sol. PA : PB = 1:2
PA = 1/3, & PB = 2l/3
K = l/3 sinθ
Or 3K = lsinθ -(1)
h = 2l/3 cosθ
3h/2 = lcosθ - (ii)
Squaring and adding eqyation (i) & (ii)

Then 9k2 + (9h2)2 = l2

4
2 2 2
9h + 36k = l

9x2 + 36y2 = l2

Ex. The eccentricity of an ellipse x2 + y2 = 1 whose latn rectum is half of its major axis is.
a 2 b2
x 2 + y2 = 1 ab b2 = a2(1-e2)
a2 b2
e2= 1- b2 = a2 – b2 b2 = 1-e2
a2 a2 a2
given 2b2 = ½ (2a)  e2 = 1- ½ = 1/2
a
2
b =1
a2 2 e=1
2
Position of a point with respect to an ellipse
Given
S= x2 + y2 - 1 pt p = (x1 , y1)
A2 b 2
S1 = x12 + y12 -1
a2 b2

(1) if s10, point lie on side

(ii) if s1 = 0, point p lie on the curve
(iii) if s1 0, point p lie in side

Interection of a line and ellipse

Let the ellipse x2 + y2 = 1
a2 b2
& line y=mn+c
x2 + (mn + c)2 = 1
a2 b2
(a2m2 + b2)x2 + 2mc.a2.x + c2a2 – a2b2 = 0
0 = (2mca2)2 - 4 (a2m2 + b2) (c2s2 – a2b2)

(i) line intersect if D0

4m2c2a4 – 4 a4m2c2 – a4m2b2 + a2b2c2 – a2b4 0
M2c2a4 – a4m2c2 +a4b2m2 – a2b2c2 +c2b2 > 0
a2 m2 + b2 y c2
ii) line touch it a2 m2 +b2 =c2
iii) line neither touch grater net
a2m2 +b2 < c2
a2m2 +b2 > c2 a2m2 +b2 =c2 a2m2 +b2 < c2

Equation of tangent
1. Point form

X2 +y2 = 1
a2 b2
P = (x1 y1)
Eqn. of Tangent
xx1 + yy1 = 1
a2 b2

2. Slope from :-
Let x2 + b2 = 1 given sl
a2 b2
of tangent ‘m’
Eqn. of tangent

Y = mx  √a2m2 +b2

(3) Parameter from

x2 +y2 = 1 p = (a soc, b sin)
a cos x + y * b sin  = 1
 a2 b2
x cos + y sin =1
a b

Drill Exergues1
1. Find the length of magor Axis and minor axis the coordinate of foci and vertices and,
the eccentricity of the ellipse 3x2 +2y2 = 6 also. Find the Equation of the dictatrix.
2. Find the Equation of the ellipse per referred to 1ts centre whose mirror axis is equal
to the distinct between the foci and whose lotus rectum 1
3. Find the length and equation of the focal radil dram from the point (4√3,5) on the
ellipse 25x2 + 16y2 = 1600
4. if equation x2 + y2 =1 represent ellipse then regent of a 1s
10-a 4-a
5. The eccentricity of an ellipse x2 + y2 = 1 whose laths rectum is half of its major axis,
a2 b2
6. find the position of the point (4-3) relative to the ellipse 5x 2 + 7y2 = 140

Ans 1) Major axis = 2√3, minor axis = 2√2, minor axis = 2√2 e = 1, Vertices (0,√3) , toci
(0, b) equation of dive train y = 3 √3
2. x2 + 2y2 = 100 (3) x + 4√3y – 2y √3 =0
11x - 4√2 y – 24 √3 =0
4. a< 4 (5) 1 (6) point line outside
√2

Equation of chord joining (a cos, b sin) and (a cos , b sin) :-

A = (a cos, b sin)
B = (a cos, b sin)
Equation of chord AB
x cos ( - ) + y sin (+) = cos ( - )
a 2 b 2 2
Point of contact :- Let given ellipse x2 + y2 = 1 & slope of tangent = m
Then equation of tangent
y = mx  √a2m2 +b2 = (y = mx c)
let contact point 1 (x1,y1)
Ean of tangent xx1 - yy1 = 1 = 0
a2 b2

 x1/a2 = y1 /b2 = -1 _
√a m b
2 2 2

2
x1 = - a m , y =  b2
+ √a2m2 +b2 √a2 m2 +b2

Ex : For what value of  do ex the line y = x+ touch el the ellipse 9x2 + 16y2 = 144
SoL : x2 * y2 = 1 a2 = 16, b2 = 5
16 9
Y = x +, m = 1, c =
For tangent C2 = a2m2 +b2
2 = 16 * 1 + 9 = 25

 = 5
Ex : Prove that the straight line ln + my +n = 0 tranches the ellipse x2 + y2 = 1 if
a2 b2
a2l2 + b2m2 =n2
sol : ln +my +n = 0
y = -ln - n
m m
 x 2 + y2 = 1
a2 b2
(-n) 2 = a2l2 + b2
m2 m2
n 2 = a 2l 2 + b 2m 2
 a2l2 +b2m2 =n2

Ex Find the Equation of the tangent to the ellipse 3x2 + 4y2 = 12 when are perpendicular
to the line y + 2x =4
Sol. 3x2 + 4y2 = 12
y = -2x +y, polar of tangent = 1/2
2 2
x +y =1
4 3
a2 = 4 b 1= 3
Equation tangent
y = mx  √ a2m2b2
y = 1/2 x  √4 * b1/4 +3
y = x √4
2y = x  2
X – 2y  4 = 0

Equation of normal in different farm :

1. Point form : Ellipse x2 +y2 = 1
a2 b2

given point P = (x1, y1)

Equation of normal
a2x - b2y = a2 b2
x1 y1

2. Parametric form :
Ellipse x2 + y2 = 1
 a2 b2
P (a cos, b sin )
Equation of normal
a2x - b2y = a2-b2
a cos b sin
 ax sin - by cos = a2-b2

3. Slop form
Ellipse x2 + y2 = 1
a2 b2
Slope of normal “m”
y = mx  m (a2-b2)
√a2 + b2 m2

Prove that the straight line lx + my + n = 0 is normal to the ellipse x2 + y2 = 1

a2 b2
Equation of normal in parameter form
ax sin - by cos = a2-b2 →(1)
ln + my + n = 0 , s normal
Compare Ean → (1) s (2)
a sin = -b cos = -(a2-b2)
l m n
sin  = -l (a2-b2)
an
Cos = - an
l (a2-b2)
cos = -m(a2-b2)
bn
sin = bx
m(a2-b2)
sin2 + cos2 =1
b2n2 + a2n2 = (a2 –b2) 2
l2 m2 n2

If the normal at end of a latex rectum of an ellipse x2 +y2 =1 passes through one
extremity if the minor a 2 b2
axis, Show that the eccentricity of the ellipse is given by e 2 = 5-1
2
2 2
Sal. x + y = 1
a2 b2
coordinate of point P
P (ae, b2/a) the equation of normal at
P (ae, b2a) 1s
a2x - b2y = a2-b2
ae b2/e
ax - ay = a2-b2
e
this is /-ass through (0,-b) here
b – a(-b) = a2 –b2 b2 =a2(1-e2)
ab = a2 – b2 (b)2 = 1 – e2
a
2
b = 1 – (b) e2 = 1 – (b)2
a a
 1-e = e
2 2

1- e2 = e4
e4 + e2 – 1 = 0, e2 = -1  1+4
2
E = 5-1
2

Properties of eccentric Angles of the Co- normal points :

1. In, General four normal can be drawn to an ellipse from any point and if  + β + γ + δ
the eccentric angles of those four co-normal points,then
 + β + γ + δ is an add multiple of 
2. If , β, γ, δ are the eccentric angles of three points an the ellipse x2 +y2 = 1, then
normal’s at which are can current , then a2
b2
β = ( + β) + β = (β +γ) + β = (γ + ) = 0
3. Co-normal points lie on a fixed curve P (x 1 , y1) Q (x2, y2) R (x3,y3) S(x4, y4) be co-
normal point so that normal from them meat in T (h 1, k1)
 P, Q, R,S lie on
(a2-b2) xy + b2k x –a2 hy =0
 (a2-b2) xy +b2kx – a2hy = 0 this came called apollomilen rectangular Hyperide.

Pair of tangents

S = x2 + y2 = 1
a2 b2
P = (x1,y1)
Equation of pair tangents (P R and PR)
SS1 = T2
(x2 + y2 -1) ( x12 + y12 -1) = { xx1 + yy1 -1} 2
a2 b2 a2 b2 a2 b2
Chord of contact :-
S=x2 +y2 -1
A2 b2
P = (x1,y1)
Equation of chard of contact RR
T=0
xx1 + yy1 - 1=0
a2 b 2
xx1 + yy1 = 1
A2 b2

Chord bisected at a given point :

S = x2 + y2 -1 chord given mid
a2 b2

Point (x1,y1) T = S1
xx1 + yy1 -1 = x12 + y12 -1
a2 b2 a 2 b2
Pole and Polar :-

S = x2 +y2 = 1
a2 b2
Pole = 6P = (x1, y1)
Equation of Polar T = 0
xx1 + yy1 -1 = 0
a2 b2
Note : - 1) Pole of the focus is the divtron point of contact.

Pole of a given line:-

Le+ Equation of polar lx + my + = 0 and ellipse x2 + y2 = 1
a2 b2
let pole is (x1,y1)
End of polar
xx1 + yy1 – 1 =0
a2 b2
lx + my +n = 0 (1)
xx1 + yy1 -1 = 0 (11)
2 2
a b
Equation 1 & 11 identical
x1 /a2 = y1 /b2 = -1
l m 
2 2
(x1,y1) = (-a l , -b m)
n n
Proper burs of Pole and Polar :-
Prop -1 If the Polar of P (x 1y1) passes through R (x2,y2) then the polar of R (x 2,y2) goes
through P (x1,y1) and Dues points are paid to be conjugate points.

Porp- 2 If the pole a line lx + my+ n = 0 lie on the connotes line l 1x + m1y + n1 =0 then the
pole of the second line wire li on the first and such lines are paid to be conjugate lines.
Prop- 3 : Pole of a given line is Bans as point of grte of tangents at its entremets.

Diameter :- The focus of the middle points of a system of parallel ends of an ellipse is
called a diameter and the point when the diameter intersects the ellipse is called the
vertex of the diameter
S = x2 + y2 -1 =0
a2 b2

Equation of diameter
y = -b2 x
a 2m

Conjugate diameters :Two diameters are paid to be conjugate when each bisects all
chords parallel to the other if y = mx and y = m, x be two conjugate diameters of an
ellipse the
M m1 = b2
a2
Properties : 1. The eccentric angler of the ends of a pair of conjugate diameters of an
ellipse differ by a right angle.
2. The path of the equares of any to conjugate sems diameters of an ellipse is constant
and Equal to the sun of the square of the semi- axis of the ellipse.
3. The Product of the focal distance of a point on an ellipse is Equal to the Square of the
semi diameter which is conjugate to the diameter through the point.
4. The tangents at the exteremitas of a part of conjugate diameter from a parallelogram
whole area is constant and Equal to Product of axes.
5. The Polar of any point colth respect to ellipse is parallel to the diameter to the one on
which the point this. It can obtains the Equation of the chord mid point is (h.k.)
Equi.- Conjugate diameters :-
Two Conjugate diameter  are called equi.-
Conjugate if then length are Equal is such ca hc then fore
(CP)2 = (CD)2
CP = CD √ a2 +b2
2

Director Circle : - The focus of the port of Intersection of the tangents to an ellipse
x2+y2
a2 b2 =1 which are perpendicular to each other is called director c 1 = cu locu x of Pt P is
director circle
LAPA1 = 90°
Equation of director cir cu
X2 + y2 = a2 + b2
Some Standard Properties of the ellipse
1. If S be the focus and G be the point when the normal at P meet l the a α i of an
ellipse then sh = e. SP and then tangent and normal at P bisect  the eternal and
Paternal angles between the total distance of P.
2. The Focus of the feet of the per particular from the foci on any tangent to an ellipse
is the auxiliary circle.
Reflection Property of ellipse
If an in conif g linght ray possess through on touch (s) Ke the can cave side of the ellipse
then it will get reflected to W and  other focus (S)

(24)
Ex. A reg emanating from the point (-3,0) is in lident on the ellipse 16x 2 +25y2 = 400 at
the Point P with ordinate 4. Find the Equation of the reflects ray after first reflection.
Sample : for P þomt

Y=4
16x +25y2 =400
2

16 x2 + 25 * 42 =400
X =0
P = (0,4)

E2 = 1 – b2 = 1 - 16
 a2 25
e= 3
5
Form ( ae, 0) = (3, 0)
Equation of reflected ray ( yl ps)
x+y =1
3 4
 4x + 3y = 12

Ex. Show that the locus of the middle path  of chords of an ellipse which par  through
a fixed point, is another ellipse
Pul. Let P (x1, y1) be the middle point of any chug AB of the ellipse x2 + y2 = 1 then
Equation of chord AB is a2 b 2

T = S1
xx1 + yy1 - 1 x12 + y12 - 1
a2 b2 a2 b2
 xx1 + yy1 = x12 + y12
a2 b2 a2 b2
if pass through fixed point 2 (h.k.)
hx1 + ky1 = x12 + y12
a2 b 2 a2 b2
this can be written a
(x1 – h)2 + (y1 –k2)2 = 1 h2 + k2
a2 b2 4 a2 b2

hen lecu  P (x1 , y1) 1

(x – h/2)2 + (y –k/2)2 = 1 h2 + k2
a2 b2 4 a2 b 2
ea Centre = (h /2, k/2)

Ex. Determine the Equation of magor and mike a α e of the ellipse

4(x – 2y +1)2 + 9 (2x +y +2)2 = 25
4 * 5 (x- 2y +1)2 + 9 * 5 * (2x + y + 2)2 = 25
√5 √5
(x- 2y +1) + (2x + y + 2)2 = 1
2

√5 √5
5/4 5/9
X2 + Y2 = 1
a2 b2
Equation of Major A * “y” = 2 x + y +2 =0
Equation of minor A * “y” = x -2y + 1 = 0
X=0
Centre X = 0, Y = 0
2x +y +2 = 0 x -2y +1 =0
X = -1, y =0 (-1,0)
Length of latuyt rectum

DRIFT EXERCIBE 2
1. Find the common tangent of x2 + y2 =4 and 2x2 + y2 =2
2. The Equation of the tangent to the ellipse 3x 2 + y2 = 3 making Equal Interrupt on the
Apex cix.
3. If the normal at any point P on the ellipse x + y2 = 1 meter the Axes h and g
uprctively the Ph : Pg is. a2 b2
4. The Eccentric Arch of one and of c diameter of x2 + 3y2 = 3 is π/6 then the eccentens
Arch of the other end will be.
5. From the point (, 3) targets are dream to x2 + y2 = 1 and ax per perpendicular
tangents of the ellipse x2 + y2 = 1 1l
9 4

Ans.
1. Ellipse completely line 15 side circle
2. y = x  2 (3) b2 : a2 (4) -5π
6
5. 2 6. x2 + y2 = 13