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Question

If α,β are the eccentric angles of the extremities of a focal chord of an ellipse, then eccentricity of the ellipse is

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Solution

Let a,b are the length of semi-major axis and semi-minor axis respectively, then (acosα,bsinα),(acosβ,bsinβ),(ae,0) are collinear.

b(sinβsinα)a(cosβcosα)=bsinα0acosαae(cosαe)(sinβsinα)=sinα(cosβcosα)e=cosα(sinβsinα)sinα(cosβcosα)sinβsinαe=sin(αβ)sinαsinβe=sin(αβ)(sinα+sinβ)sin2αsin2βe=sinα+sinβsin(α+β)[sin(A+B)sin(AB)=sin2Asin2B]

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