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Q.

Let a,b and λ be positive real numbers. Suppose P is an end point of the latus rectum of the parabola y2=4λx, and suppose the ellipse x2a2+y2b2=1 passes through the point P . If the tangents to the parabola and the ellipse at the point P are perpendicular to each other, then the eccentricity of the ellipse is

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a

12

b

12

c

13

d

25

answer is A.

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Detailed Solution

y2=4λx,P(λ,2λ) Slope of the tangent to the parabola at point Pdydx=4λ2y=4λ22λ=1 Slope of the tangent to the ellipse at P2xa2+2yy′b2=0 As tangents are perpendicular, y′1=−1⇒y'=-1⇒2λa2−4λb2=0⇒a2b2=12 at the point λ,2λ ⇒ e=1−12=12,

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