Question

The tangent at a point P(θ) to the elllipse x2a2+y2b2=1 cuts the auxiliary circle at point Q and R. If QR subtends a right angle at the centre C of the ellipse, then the eccentricity of the ellipse is:

A
11+cos2θ
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B
11+sin2θ
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C
1+cos2θ
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D
1+sin2θ
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Solution

The correct option is D 1√1+sin2θThe equation of the tangent at P(θ) to the ellipse x2a2+y2b2=1 isxacosθ+ybsinθ=1The combined equation of the lines CQ and CR isx2+y2=a2(xacosθ+ybsinθ)2The lines represented by this equation are at right angle.∴1−cos2θ+1−a2b2sin2θ=0⇒sin2θ+1−11−e2sin2θ=0⇒1−e2=sin2θ1+sin2θ⇒e2=11+sin2θ⇒e=1√1+sin2θ

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