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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

A
cosα+cosβcos(α+β)
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B
sinαsinβsin(αβ)
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C
cosαcosβcos(αβ)
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D
sinα+sinβsin(α+β)
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Solution

## The correct option is D sinα+sinβsin(α+β)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(sinβ−sinα)a(cosβ−cosα)=bsinα−0acosα−ae⇒(cosα−e)(sinβ−sinα)=sinα(cosβ−cosα)⇒e=cosα(sinβ−sinα)−sinα(cosβ−cosα)sinβ−sinα⇒e=sin(α−β)sinα−sinβ⇒e=sin(α−β)(sinα+sinβ)sin2α−sin2β∴e=sinα+sinβsin(α+β)[∵sin(A+B)sin(A−B)=sin2A−sin2B]

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