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If alpha and beta are the eccentric an...

If `alpha and beta` are the eccentric angles of the extremities of a focal chord of an ellipse, then prove that the eccentricity of the ellipse is `(sinalpha+sinbeta)/("sin"(alpha+beta))`

Text Solution

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Equation of chord through points `P(a cos alpha, sin alpha), Q(a cos beta, b sin beta)` is `(x)/(a) sin. (alpha+beta)/(2)+(y)/(b) sin. (alpha+beta)/(2) = cos. (alpha-beta)/(2)`
If it passes through focus, S(ae,0) then
`ecos.(alpha+beta)/(2)=cos.(alpha-beta)/(2)`
`rArre=(cos.(alpha-beta)/(2))/(cos.(alpha+beta)/(2))`
`=(2sin.((alpha+beta)/(2))*cos.((alpha-beta)/(2)))/(2sin((alpha+beta)/(2))*cos((alpha+beta)/(2)))=(sinalpha+sinbeta)/(sin(alpha+beta))`
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