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

A
(±352,±27)
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B
(±352,±194)
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C
(±23,±17)
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D
(±23,±437)
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Solution

## The correct option is C (±2√3,±17)Normal to the ellipse at any point ′ϕ′ is, 4xsecϕ−2ycosecϕ=12Q≡(3cosϕ,0)Let M≡(α,β)∴α=3cosϕ+4cosϕ2=72cosϕ⇒cosϕ=27αand β=sinϕNow using, cos2ϕ+sin2ϕ=1⇒449α2+β2=1⇒449x2+y2=1⇒ latus rectum x=±2√34849+y2=1⇒y=±17Thus required point is (±2√3,±17)

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