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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 mid-point of the line segment PQ, then the locus of M intersects the latus rectum 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 (±23, ±17)
EquationofnormalatpointPis:axcosθbysinθ=(ae)2Xintercept:X=(ae)2cosθaLetM(h,k)bethemidpointofPQthenh=acosθ(1+e2)2andk=bsinθ2Or,cosθ=2ha(1+e2)andsinθ=2kbSquaringandaddingweget:4h2(a(1+e2))2+4k2b2=1a=4b=2e=324h249+k21=1(1)Equationoflatusrectumisx=±ae=±23Pointofintersectionwithequation1is:41249+k21=1or,k2=149k=±17pointsare(±23,±17)

Option [C]

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