Let a , b and λ be positive real numbers. Suppose P is an end point of the latus rectum of the parabola y 2 = 4 λ x , and suppose the ellipse x 2 a 2 + y 2 b 2 = 1 passes through the point P . If the tangents to the parabola and t

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Let a , b and &#955; be positive real numbers. Suppose P is an end point of the latus rectum of the parabola y 2 = 4 &#955; x , and suppose the ellipse x 2 a 2 + y 2 b 2 = 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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Correct answer: A. 1 2

y 2 = 4 λ x ⇒ d y d x P = 1 = m 1 Also, x 2 a 2 + y 2 b 2 = 1 ⇒ d y d x P = − b 2 2 a 2 = m 2 Given both tangents are perpendicular. ⇒ m 1 ⋅ m 2 = − 1 ⇒ b 2 = 2 a 2 And a 2 = b 2 1 − e 2 ⇒ 1 = 2 1 − e 2 ⇒ e = 1 2

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