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Question

If the tangent at θ on the ellipse x2a2+y2b2=1 meets the auxiliary circle at two points which subtend a right angle at the centre, then e2(2cos2θ)=

A
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B
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C
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Solution

The correct option is A 1
Homogenising the equation of the auxiliary circle x2+y2=a2 with the help of the tangent at θ
i.e., xacosθ+ybsinθ=1, we get x2+y2a2(xacosθ+ybsinθ)2=0
Since this equation represents a pair of perpendicular, lines, we have sum of the coefficients of x2 and y2 is zero
(1cos2θ)+1a2b2sin2θ=0b2sin2θ+b2a2sin2θ=0a2(1e2)(1+sin2θ)a2sin2θ=0
(1e2)(1+sin2θ)=sin2θ
On simplification, we get, e2(2cos2θ)=1

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