Download Ellipse P

Pure Maths/Ellipse/p.1
Ellipse
For the standard form of the equation of the ellipse, the focus is (ae , 0) and the directrix is x =
Let P(x, y) be a variable point on the ellipse.
y
Distance of (x, y) from the focus (ae, 0)
a
Distance of (x, y) from the directrix x =
e
or
= e
Distance of (x, y) from the focus (- ae, 0)
a
Distance of (x, y) from the directrix x = e
(-a, 0)
x
x
= e

 x - ae



a 
+ y2 = e2  - x
e 
2



x2 1 - e 2 + y2 = a 2 1 - e 2
x2
y2
+
a2
a 2 1 - e2

= 1

a
e

A’
(-a,0)
(a, 0)
x
x
F(ae, 0)
Principal
focus
x=a
e
x=a
y
e
B(0,b)
x2
y2
+
= 1
2
2
a
b
where b2 = a 2 1 - e 2
x
a
e
x=x=-

0
F’(-ae,0)
Secondary
focus
where 0 < e < 1.
2
a
.
e
x
F’(-ae,0)
P
x
A(a, 0)
x
F(ae, 0)
x
.
B’(0,-b)
  a
a 
Consider PF + PF’ = e  - x  e  x -  -   = 2a
e 
  e
∴The locus of a variable point moving in such a way that the sum of its distances from 2 fixed points
(foci) is a constant is an ellipse.
Remarks
1.
A, A’, B, B’ - vertices of the ellipse ;
AA’ - major axis
x2
y2
+
= 1
32
22
origin 0
BB’ - minor axis
;
y
- centre of the ellipse
y
x2
y2
+
= 1
22
32
x
x
Pure Maths/Ellipse/p.2
2.
The chord through the focus and perpendicular to the major axis
- latus rectum (rectara)
y
(ae)2
y2
+
= 1
a2
b2

L

b4
 y2 = b 2 1 - e 2 =
a2
∴
3.
2b 2
length of latus rectum =
LL' 2b2
1
=

=
AF
a a(1 - e)
x
F(ae, 0)
a

2a 2 1 - e2
a 2 1 - e

A
x
L’
= 2 (1 + e) < 4
Point of intersection of the major axis and minor axis is called the centre of the ellipse.
Any chord through the centre is called a diameter . One half of a diameter is called a semi-diameter .
4.
The circle described on the major axis of an ellipse as diameter is known as the auxiliary circle of the
ellipse. x 2 + y 2 = a 2
5.
y
Parametric Equation :
(a) Trigonometric Parametric Equations
auxiliary circle :
x2 + y2 = a 2
a
x2
y2
+
= 1
a2
b2

 x = a cos 
 
 y = b sin 
x
P
0
where  is the eccentric angle and 0   < 2 .
(b) Algebraic Parametric Equations
 x = a cos 

 y = b sin 
Let

t = tan

.
2
 
a 1 - t2
x =
1 + t2
;
y =
2bt
1 + t2
Example
1.
x2
y2
A circle cuts the ellipse
+
= 1 at the points A, B, C and D.
a2
b2
Show that AB, CD are equally inclined to the x-axis.
Let the equation of AB be y - mx - c = 0 and the equation of CD be y - m’x - c’ = 0.
x
Pure Maths/Ellipse/p.3
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2.
x2
y2
+
= 1 . Variable line through R is drawn
2
2
a
b
y
meeting the ellipse at two points P and Q.
x R(h. k)
Find the locus of the mid-point of the chord PQ.
Let R(h, k) be an external point of the ellipse
P
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x
Q
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Pure Maths/Ellipse/p.4
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Classwork (91)
Chords with slope equal to 1 are drawn in the ellipse
2x2 + 2xy + y2 = 1 .
Prove that the mid-points of those chords lie on a straight line, and find the equation of this line.
(Answer : 3x + 2y = 0 )
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Tangents of an Ellipse
The equation of tangent to the ellipse
x2
y2
+
= 1 at the point (x1 , y1 ) or (a cos  , b sin  ) is
a2
b2
xx1
yy
+ 1 = 1
a2
b2
or
x cos
y sin
+
= 1 .
a
b
Given a line y = mx + c , what is the condition for the line to be a tangent to the ellipse
Note : If the line is vertical, e.g. x = c for some constant c,
x2
y2
+
= 1 ?
a2
b2
Pure Maths/Ellipse/p.5
when c =  a , it is a tangent to the ellipse
when
c
> a , no intersection
when
c
< a , two distinct points of intersection





x2
y2
+
= 1
a2
b2
y = mx + c

b2 x2 + a 2(mx + c)2 = b2a 2

b2 + a 2m2  x2 + 2a 2mcx + a 2c2 - b2a 2  = 0
The line y = mx + c is a tangent to the ellipse
∴

x2
y2
+
= 1
a2
b2

iff
△ = 0


4a 4m 2c2 - 4 b2 + a 2m 2 a 2c2 - b2a 2 = 0

a 2 m 2 c2 - b 2 c 2 + a 2 m 2 c2 - b 4 - a 2 b 2 m 2 = 0

b 2 c2 - b 2 - a 2 m 2 = 0

c2 = b2 + a 2m2




c2 = b 2 + a 2 m 2
tangent to the ellipse
 b
 0
is a necessary and sufficient condition for the line y = mx + c to be a t
x2
y2
+
= 1.
a2
b2
Hence, for any given value of m, the equations of tangents are
y = mx 
b2 + a 2 m 2 .
Example
1.
Find the equation of the locus of a variable point P moving
on the xy plane in such a way that the two tangents drawn
from P to the ellipse
x2
y2
+
= 1 are perpendicular.
a2
b2
y
P
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Pure Maths/Ellipse/p.6
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2.
Find the equation of the common tangents to the ellipse x2 + 2y2 = 6 and the circle 2x2 + 2y2 = 9 .
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3. Let P and Q be two points on the ellipse
x2
y2
+
= 1 with eccentric angles  and 
a2
b2
respectively. Show that, if  -  is a constant, then the locus of the mid-point of the chord PQ is also
an ellipse.
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Pure Maths/Ellipse/p.7
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4.
If from an external point T(h, k), tangents TP and TQ are
y
x2
y2
drawn to the ellipse
+
= 1 , show that the area of
a2
b2
the quadrilateral OPTQ is
T(h, k)
b2h 2 + a 2k 2 - a 2b2 .
P
Q
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Note : Equation of normal at the point (x1 , y1 ) to the ellipse
b2 x1
dy
= d x (x , y )
a 2 y1
1 1

x2
y2
+
= 1 :
2
2
a
b
Equation of mormal at (x1 , y1 ) :
a 2 y1
y - y1 =
x - x1
b2 x1



a2 x
b2 y
= a 2 - b2
x1
y1
If (x1 , y1 ) is expressed as (a cos  , b sin  ), then the equation of the normal is :
x
Pure Maths/Ellipse/p.8
ax
cos 
by
= a 2 - b2
sin 
where cos   0 and sin   0
Diameter and Conjugate Diameters
y
Equation of the chord with T(h, k) as the mid-point :
r1
 x = h + r cos

 y = k + r sin

 h + r cos 
2
+
a2
 k + r sin 
r2
2
b2
= 1
For T to be the mid-point of the chord, r1 = - r2 .
sum of roots = 0


2b2h cos + 2a 2k sin
b2 cos2 + a 2 sin 2

2b2 h cos = - 2a 2 k sin

tan = -
= 0
b2 h
a 2k
Equation of the chord with (h, k) as mid-point :
y-k =-
b2 h
 x - h
a 2k
a 2k (y - k) = - b2h  x - h
a 2 k y + b2 h x = a 2 k 2 + b2 h 2
OR
hx
ky
h2
k2
+
=
+
a2
b2
a2
b2
b2 h
where the slope of the chord = a 2k
If m is used to denote the slope, then m = -
x T(h,k)
x
with roots r1 and r2
2h cos
2k sin
+
2
a
b2
= 0
cos2 
sin 2 
+
a2
b2
x2
y2
+
= 1
2
2
a
b
b2 h
.
a 2k
Pure Maths/Ellipse/p.9

∴
k = -
b2
h
a 2m
y = -
b2
x
a 2m
y
The mid-points of parallel chords with common slope m
lie on the straight line y = -
b2
x .
a 2m
x
The mid-points of a system of parallel chords with common
slope m on a line with slope -
b2
and passing through the centre of
a 2m
the ellipse. In other words, the mid-points lie on a diameter.
The diameters are said to be conjugate to one another if each of them
bisects the chords parallel to the other.
 y = mx

2
 y = - b x

a 2m

The product of the slopes of two conjugate diameters is -
slope of OC =
b
tan
a
slope of OD =
b
tan
a
b2
.
a2
y
D(a cos  , b sin  )
C(a cos  , b sin  )
For CC’ and DD’ to be conjugate diameters,
b2
b2
tan tan = a2
a2

As
∴
tan tan =
tan ( -  ) =
-1
tan - tan
,
1 + tan tan
 - = 

2
If CC’ = DD’, then they are called equiconjuagte diameters.
O
x
D’
C’
Pure Maths/Ellipse/p.10
Example
1.
If OP and OQ are conjugate semi-diameters of the ellipse
point P from OQ, show that OD =
x2
y2
+
= 1 , p is the distance of the
a2
b2
ab
and that the mid-point of the chord PQ lies on the ellipse
p
x2
y2
1
.
+
=
2
2
2
a
b
y
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Q
P
p
D
O
x
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2.(1982)
Given an ellipse (E) :
x2
y2
+
= 1.
a2
b2
(a) P is a point (h, k). If y = mx + c is a tangent drawn from P to (E), show that
a 2 - h 2 m2 + 2hkm + b2 - k 2  = 0 .
(b) Two non-parallel lines of slopes m1 and m2 are equally inclined to the line y = nx.
Show that
1 - n 2 m1 + m2 + 2n m1m2 -1= 0 .
(c) Tangents of (E) equally inclined to the line y = nx
intersect at the point P. Using the results of (a)
and (b), or otherwise, find the equation of the locus of the point P.
Pure Maths/Ellipse/p.11
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P.M./Ellipse/p.11
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1 - 2
y
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2
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1
O
3
4
y = nx
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(c)
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W.42 Q. 2 , 3 , 4
H.W. W.42 Q 5 - 7 W.43 Ex.B Q. 4 , 5, 11
END
x