## Ellipse Questions and Answers Part-1

1. The locus of the points of intersection of the tangents at the extremities of the chords of the ellipse $x^{2}+2y^{2}=6$    which touch the ellipse $x^{2}+4y^{2}=4$    is
a) $x^{2}+y^{2}=4$
b) $x^{2}+y^{2}=6$
c) $x^{2}+y^{2}=9$
d) none of these

Explanation:

2. The normal at an end of a latus rectum of the ellipse $x^{2}/a^{2}+y^{2}/b^{2}=1$    passes through an end of the minor axis if
a) $e^{4}+e^{2}=1$
b) $e^{3}+e^{2}=1$
c) $e^{2}+e=1$
d) $e^{3}+e =1$

Explanation:

3. If an ellipse sides between two perpendicular lines, then the locus of the centre is
a) a parabola
b) an ellipse
c) a circle
d) none of these

Explanation: Let 2a, 2b be the length of the major and minor axes respectively of the ellipse If the ellipse slides between two perpendicular lines, the point of intersection P of these lines being the point of intersection of perpendicular tangents lies on the Director circle of the ellipse

4. If the tangent at a point (a cos $\theta$ , b sin $\theta$ ) on the ellipse $x^{2}/a^{2}+y^{2}/b^{2}=1$    meets the auxiliary circle in two points, the chord joining them subtends a right angle at the centre; then the eccentricity of the ellipse is given by
a) $\left(1+\cos^{2}\theta\right)^{-1/2}$
b) $1+\sin^{2}\theta$
c) $\left(1+\sin^{2}\theta\right)^{-1/2}$
d) $1+\cos^{2}\theta$

Explanation: Equation of the tangent (a cos $\theta$ , b sin $\theta$ ) to the ellipse x2 /a2 + y2 /b2 = 1 is

5. If $F_{1}=\left(3,0\right),F_{2}=\left(-3,0\right)$      and P is any point on the curve $16x^{2}+25y^{2}=400$   , then $PF_{1}+PF_{2}$   equals
a) 8
b) 6
c) 10
d) 12

Explanation:

6. A common tangent to the circle $\left(x-6\right)^{2}+y^{2}=4$   , parabola $y^{2}=x-4$   and the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$
is
a) x = –4
b) x = 4
c) y = –3
d) y = 3

Explanation: From the figure, it is clear that a common

7. AB is a variable chord of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1.$     if AB subtends a right angle at the origin, and $\frac{1}{OA^{2}}+\frac{1}{OB^{2}}=\frac{3}{a^{2}}$
then eccentricity of the ellipse is
a) $\frac{1}{\sqrt{3}}$
b) $\frac{1}{\sqrt{2}}$
c) $\frac{1}{2}$
d) $\frac{1}{3}$

Explanation: Suppose OB makes an angle $\theta$ with the positive direction of the x-axis, then OA makes an angle of

8. Suppose the ellipse $x^{2}+\frac{y^{2}}{a^{2}}=1$    intersects the ellipse $\frac{x^{2}}{4^{2}}+y^{2}=1$   , in four distirict point. If $a=b^{2}-5b+7$    then b cannot lie in the set
a) (–1, 1)
b) (1, 4)
c) (2, 3)
d) (–1, 3)

Explanation: As the ellipse intersect in four distinct points, a > 1

9. The circle $3x^{2}+3y^{2}=a^{2}+3b^{2}$     meets the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$      (a > b ) in exactly two points. The eccentricity of the ellipse is
a) $\frac{1}{\sqrt{3}}$
b) $\sqrt{\frac{2}{3}}$
c) $\frac{1}{\sqrt{2}}$
d) $\frac{1}{3}$

10. If y = mx + c is a normal to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$    , then $c^{2}$  equals
a) $\frac{\left(a^{2}-b^{2}\right)^{2}}{a^{2}m^{2}}$
b) $\frac{\left(a^{2}-b^{2}\right)^{2}m^{2}}{a^{2}m^{2}+b^{2}}$
c) $\frac{\left(a^{2}-b^{2}\right)^{2}}{a^{2}+b^{2}m^{2}}$
d) $\frac{\left(a^{2}-b^{2}\right)^{2}m^{2}}{a^{2}+b^{2}m^{2}}$