## Ellipse Questions and Answers Part-4

1. If the chords of contact of tangents from two points $\left(x_{1},y_{1}\right)$   and $\left(x_{2},y_{2}\right)$   to the ellipse $x^{2}/a^{2}+y^{2}/b^{2}=1$    are at right angles then $x_{1}x_{2}/y_{1}y_{2}$   is equal to
a) $a^{2}/b^{2}$
b) $-b^{2}/a^{2}$
c) $-a^{4}/b^{4}$
d) $-b^{4}/a^{4}$

Explanation:

2. Let $E_{1}$  be the ellipse $\frac{x^{2}}{a^{2}+2}+\frac{y^{2}}{b^{2}}=1$     and $E_{2}$  be the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}+1}=1.$    The number of points from which to perpendicular tangents can be drawn to each of $E_{1}$   and $E_{2}$  is …
a) 0
b) 1
c) 2
d) 4

Explanation:

3.An equation of the ellipse centered at (0,0) having eccentricity $\frac{3}{5}$ and passing through (4, 0) is
a) $16x^{2}+25y^{2}=256$
b) $25x^{2}+16y^{2}=400$
c) $25x^{2}+16y^{2}=256$
d) $16x^{2}+25y^{2}=400$

Explanation:

4. The number of value of c for which y = 5x + c is a tangent to the ellipse $\frac{x^{2}}{25}+y^{2}=1$    is
a) 1
b) 2
c) 4
d) 6

Explanation:

5. If the equation $\frac{x^{2}}{10-2a}+\frac{y^{2}}{4-2a}=1$     represents an ellipse, then 'a' lies in the interval
a) $\left(-\infty ,5\right)$
b) (2, 5)
c) $\left(-\infty ,2\right)$
d) $\left(5,\infty \right)$

Explanation:

6. If $\left(\tan\theta_{1} \right)\left(\tan\theta_{2} \right)=\frac{-a^{2}}{b^{2}},$
then the chord joining two points $P_{1}\left(\theta_{1} \right)$   and $P_{2}\left(\theta_{2} \right)$   on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$    will subtend a right angle at
a) focus (ae, 0)
b) focus (–ae, 0)
c) centre (0,0)
d) vertex (a,0))

Explanation:

7.Let $P\left(a \cos\theta ,b\sin\theta\right)$    and $Q\left(a \cos\phi ,b\sin\phi\right)$      where $\theta+\phi=\frac{\pi}{2}$     be two points on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$
The locus of point of intersections of normals at P and Q is
a) ax + by = 0
b) ax - by = 0
c) x + y = 0
d) x + y = a + b

Explanation:

8. The locus of the point of intersection of the tangents at the extremities of the chord of the ellipse $x^{2}+2y^{2}=6$    which touches the ellipse $x^{2}+4y^{2}=4$    is
a) $x^{2}+y^{2}=6$
b) $x^{2}+y^{2}=2$
c) $x^{2}+y^{2}=9$
d)$x^{2}+y^{2}=12$

Explanation:

9. If P is a point on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=$    with foci S and S' and eccentricity e, then locus of the incentre of the triangle PSS' is an ellipse of eccentricity
a) $\sqrt{\frac{1-e}{1+e}}$
b) $\sqrt{\frac{e}{1+e}}$
c) $\sqrt{\frac{2e}{1+e}}$
d) $\sqrt{\frac{1-2e}{1+e}}$

10. Equation of a tangent to the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$    which cuts off equal intercepts on the axes is
a) $x+y-\sqrt{41}=0$
b) $x-y+\sqrt{41}=0$