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 (±2√3, ±17)EquationofnormalatpointPis:axcosθ−bysinθ=(ae)2X−intercept:X=(ae)2∗cosθaLetM(h,k)bethemid−pointofPQthenh=acosθ(1+e2)2andk=bsinθ2Or,cosθ=2ha(1+e2)andsinθ=2kbSquaringandaddingweget:4h2(a(1+e2))2+4k2b2=1a=4b=2e=√32∴4h249+k21=1−−−−−−−−−−(1)Equationoflatusrectumisx=±ae=±2√3Pointofintersectionwithequation1is:4∗1249+k21=1or,k2=149k=±17∴pointsare(±2√3,±17)Option [C]

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