EllipseHard

Question

The normal at a point P on the ellipse x2 + 4y2 = 16 meets the x-axis at Q. If M is the mid point of the line segment PQ, then the locus of M intersects the latus rectums of the given ellipse at the points

Options

A.
B.
C.
D.

Solution

         
Normal is 4x sec φ - 2y cosec φ = 12
Q ≡ (3 cos φ, 0)
M ≡ (α, β)
α = cos φ
⇒ cos φ = α
β = sin φ
cos2φ + sin2 φ = 1
⇒ α2 + β2 = 1 ⇒ x2 + y2 = 1
⇒ latus rectum x = ± 2√3
+ y2 = 1 ⇒ y = ±
(± 2√3, ± 1/7).

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