ELLIPSE
Q.1
PG is the normal to a standard ellipse at P, G being on the major axis. GP is produced outwards to Q so
2
2
that PQ = GP. Show that the locus of Q is an ellipse whose eccentricity is a b & find the equation
a 2 b2
of the locus of the intersection of the tangents at P & Q.
Q.2
P & Q are the corresponding points on a standard ellipse & its auxiliary circle. The tangent at P to the
ellipse meets the major axis in T. Prove that QT touches the auxiliary circle.
Q.3
The point P on the ellipse
Q.4
Q.5
Q.6
Q.7
Q.8
is joined to the ends A, A of the major axis. If the lines through
P perpendicular to PA, PA meet the major axis in Q and R then prove that
l(QR) = length of latus rectum.
x 2 y2
Let S and S' are the foci, SL the semilatus rectum of the ellipse 2 2 1 and LS' produced cuts the
b
a
(1 e 2 )
a , where 2a is the length
ellipse at P, show that the length of the ordinate of the ordinate of P is
1 3e 2
of the major axis and e is the eccentricity of the ellipse.
x 2 y2
1 touches at the point P on it in the first quadrant & meets the
a 2 b2
coordinate axis in A & B respectively. If P divides AB in the ratio 3 : 1 find the equation of the tangent.
A tangent to the ellipse
x 2 y2
1 (a > b) & QCQ is the corresponding diameter of the
a 2 b2
x 2 y 2 formed by the tangent at P, P', Q & Q' is
auxiliary circle, show that the area of the parallelogram
1
a 2 b2
8a 2 b
where is the eccentric angle of the point P.
(a b) sin 2
x 2 y2
If the normal at the point P() to the ellipse
1 , intersects it again at the point Q(2),
14 5
show that cos = (2/3).
PCP is a diameter of an ellipse
x 2 y2
A normal chord to an ellipse 2 2 1 makes an angle of 45 with the axis. Prove that the square of
b
a
32a 4 b 4
its length is equal to 2
(a b 2 ) 3
x 2 y2
1 , the tangents at which meet in
a 2 b2
(h, k) & the normals in (p, q), prove that a2p = e2hx1 x2 and b4q = e2k y1y2a2 where 'e' is the eccentricity.
Q.9
If (x1, y1) & (x2, y2) are two points on the ellipse
Q.10
A normal inclined at 45 to the axis of the ellipse
x 2 y2
1 is drawn. It meets the x-axis & the y-axis in P
a 2 b2
& Q respectively. If C is the centre of the ellipse, show that the area of triangle CPQ is
Q.11
Tangents are drawn to the ellipse
(a 2 b 2 ) 2
sq. units.
2(a 2 b 2 )
a2
x 2 y2
2
2
from
the
point
. Prove that they
,
a
1
2
2
a 2 b2
a b
intercept on the ordinate through the nearer focus a distance equal to the major axis.
Q.12
P and Q are the points on the ellipse
x 2 y2
1 . If the chord P and Q touches the ellipse
a 2 b2
4 x 2 y 2 4x
0 , prove that sec + sec = 2 where , are the eccentric angles of the points P and Q.
a 2 b2 a
x 2 y2
2 1 & the circle x2 + y2 = r2 ; where a > r > b.
2
a
b
A focal chord of the ellipse, parallel to AB intersects the circle in P & Q, find the length of the perpendicular
drawn from the centre of the ellipse to PQ. Hence show that PQ = 2b.
Q.13
A straight line AB touches the ellipse
Q.14
Show that the area of a sector of the standard ellipse in the first quadrant between the major axis and a
line drawn through the focus is equal to 1/2 ab ( e sin ) sq. units, where is the eccentric angle of the
point to which the line is drawn through the focus & e is the eccentricity of the ellipse.
Q.15
A ray emanating from the point ( 4, 0) is incident on the ellipse 9x2 + 25y2 = 225 at the point P with
abscissa 3. Find the equation of the reflected ray after first reflection.
Q.16
If p is the length of the perpendicular from the focus S of the ellipse
then show that
b2
2a
1.
p 2 (SP )
x 2 y2
1 on any tangent at 'P',
a 2 b2
x 2 y2
Q.17 In an ellipse 2 2 1, n1 and n2 are the lengths of two perpendicular normals terminated at the major
b
a
axis then prove that :
1
1
a 2 b2
=
n12 n 22
b4
x 2 y2
2 1
2
2 a2
b
x
y
Q.18 If the tangent at any point of an ellipse 2 2 1 makes an angle with the major axis and an angle
b
a
with the focal radius of the point of contact then show that the eccentricity 'e' of the ellipse is given by
cos
the absolute value of
.
cos
Q.19
Using the fact that the product of the perpendiculars from either foci of an ellipse
upon a
tangent is b2, deduce the following loci. An ellipse with 'a' & 'b' as the lengths of its semi axes slides
between two given straight lines at right angles to one another. Show that the locus of its centre is a circle
& the locus of its foci is the curve, (x2 + y2) (x2 y2 + b4) = 4 a2 x2 y2.
Q.20
x 2 y2
1 intercept on the x-axis a constant length c, prove that
a 2 b2
the locus of the point of intersection of tangents is the curve
4y2 (b2x2 + a2y2 a2b2) = c2 (y2 b2)2.
If tangents are drawn to the ellipse
ANSWER KEY
Q.1 (a2 b2)2 x2y2 = a2 (a2 + b2)2 y2 + 4 b6x2
Q.13
r 2 b2
Q.5 bx + a 3 y = 2ab
Q.15 12 x + 5 y = 48 ; 12 x 5 y = 48
