Question

# The tangent at any point P on the ellipse x2a2+y2b2=1 meets the auxiliary circle in two points which subtend right angle at the centre of the ellipse. If θ be the eccentric angle of P and e be the eccentricity of the ellipse, then sinθ=

A
1e2
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B
e/1e2
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C
1e2/e
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D
None of these
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Solution

## The correct option is C √1−e2/e Equation of the tangent at P(θ) on the given ellipse is xcosθa+ysinθb=1⋯(1) Equation of the auxiliary circle of the given ellipse is x2+y2=a2⋯(2) If A,B be the intersection points of the above tangent and the auxiliary circle, then equation of the pair of straight lines OA,OB (O is the origin) is x2+y2−a2(xcosθa+ysinθb)2=0⋯(3) According to the given condition, equation (3) must represent a pair of perpendicular straight lines. Therefore, we have coeff. of x2+ coeff. of y2=0 i.e. (1−cos2θ)+(1−a2b2sin2θ)=0 i.e. sin2θ(a2b2−1)=1 i.e. sin2θ=(b2a2−b2)=1−e2e2 i.e. sinθ=√1−e2e

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