The normal at a variable point P on ellipse x2a2+y2b2=1, (a > b) meets axes of the ellipse in Q and R. Then which of the following is/are correct statement(s)

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the locus of mid-point of QR is ellipse

the locus of mid-point of QR is conic whose eccentricity is same as of given ellipse

The locus of mid-point of QR is hyperbola

The locus of mid-point of QR is conic with its eccentricity half of the eccentricity of given ellipse

answer is A.

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Detailed Solution

Equation of Normal to the Ellipse x2a2+y2b2=1 at P(acosθ,bsinθ) is a2xacosθ−b2ybsinθ=a2-b2axcosθ−bysinθ=a2-b2 it cuts x -aris at QQ=a2-b2cosθa,0 and Normal cuts y -axis at RR0,-a2-b2sinθb Let x1,y1 be the mid point of QR Hence x1=a2-b2cosθ2a,y1:-a2-b2sinθ2bcosθ=2ax1a2-b2 sinθ=-2by1a2-b2 Eliminate θ ,4a2x12a2-b22+4b2y12a2-b22=1x12a2-b224a2+y12a2-b224b2=1 The locus of mid point is x2a4e44a2+y2a4e44b2=1 This represents Ellipse Eccentricity e=1-1a21b2=1-b2a2 Same as that of given Ellipse

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