If a variable tangent of the circle x2+y2=1 intersect the ellipse x2+2y2=4 at P

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Question

If a variable tangent of the circle

x^{2} + y^{2} = 1

intersect the ellipse

x^{2} + 2 y^{2} = 4

at P and Q then the locus of the points of intersection of the tangents at P and Q is

Options:

A . A circle of radius 2 units

B . A parabola with fouc as (2, 3)

C . An ellipse with eccentricity √34

D . An ellipse with length of latus rectrum is 2 units

Answer: Option D
:
D

x^{2} + y^{2} = 1

;

x^{2} + 2 y^{2} = 4

Let

R (x_{1}, y_{1})

is point of intersection of tangents drawn at P,Q to ellipse

\Rightarrow

PQ is chord of contact of

R (x_{1}, y_{1})

\Rightarrow x x_{1} + 2 y y_{1} - 4 = 0

This touches circle

\Rightarrow r^{2} (l^{2} + m^{2}) = n^{2} \Rightarrow 1 (x_{1}^{2} + 4 y_{1}^{2}) = 16 \Rightarrow x^{2} + 4 y^{2} = 16 i s e l l i p s e w i t h e c c e n t r i c i t y e = \frac{\sqrt{3}}{2} a n d l e n g t h o f l a t u s r e c t u m L L^{1} = 2

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