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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

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An ellipse with length of latus rectum is 2 units

A circle of radius 2 units

An ellipse with eccentricity

34

A parabola with focus as (2,3)

answer is D.

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Detailed Solution

x 2 + y 2 = 1 : x 2 + 2 y 2 = 4

Let

R x 1, y 1

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

⇒ P Q

is chord of contact of

R x 1, y 1 ⇒ x x 1 + 2 y y 1 − 4 = 0

This touches circle

⇒ r 2 l 2 + m 2 = n 2 ⇒ 1 x 12 + 4 y 12 = 16

⇒ x 2 + 4 y 2 = 16

is ellipse

e = 32 : L L ′ = 2

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