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Question

# The normal at P(θ) and D(θ+π2) meet the major axis of x2a2+y2b2=1 at Q and R. Then PQ2+DR2=

A
b2(1e2)(2e)2
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B
a2(1e2)(2e)2
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C
a2(1+e2)(2+e2)
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D
b2(1+e2)(2+e2)
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Solution

## The correct option is B a2(1−e2)(2−e)2Equation of the normal at P(θ) is axcosθ−bysinθ=a2e2 This normal meets the major axis at Q = (ae2cosθ,0) Equation of the normal at D(π2+θ) ie D(−asinθ,bcosθ) is −axsinθ−bycosθ=a2e2 This normal meets the major axis at R =(−ae2sinθ,0) ∴PQ2+DR2=[(ae2cosθ−acosθ)2+b2sin2θ]+[(ae2sinθ−asinθ)2+b2cos2θ] =a2(1−e2)(2−e2) [on simplification]

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