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Q.

Let $P_{1}$ and $P_{2}$ be the feet of perpendiculars from foci of ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1$ upon any tangent at $P$ on the ellipse. Now tangents at $P_{1}$ and $P_{2}$ to the auxilliary circle meets at $Q .$ Then eccentricity of the locus of the point $Q$ is

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a

0.6

b

1

c

0.8

d

$\sqrt{2}$

answer is D.

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Detailed Solution

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Let $Q (h, k)$
Equation of chord of contact to auxiliary circle $\frac{h x}{a^{2}} + \frac{b y}{a^{2}} = 1$
Equation of tangent to ellipse $\frac{x \cos θ}{a} + \frac{y \sin θ}{b} = 1$
Comparing $h = a \cos θ, k = \frac{a^{2}}{b} \sin θ$
So clearly locus is an ellipse of same eccentricity.

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