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Question:

The normal at a point P on the ellipse x² + 4y² = 16 meets the x-axis at Q. If M is the midpoint of the line segment PQ, then the locus of M intersects the latus rectum of the given ellipse at the points (±3√5/2, ±2/7)

(±3√5/2, ±2/7)

(±2√3, ±1/7)

(±2√3, ±4√37)

(±3√5/2, ±√19/4)

Solution:

Normal to the ellipse at any point φ is, 4xsecφ - 6ycosecφ = 12
Q = (3cosφ, 0)
Let M = (α, β) ⇒ α = 3cosφ + 4cosφ/2 = 7/2cosφ ⇒ cosφ = 2/7α and β = sinφ
Now using, cos²φ + sin²φ = 1 ⇒ 4/49α² + β² = 1 ⇒ 4/49x² + y² = 1