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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))`

A

`(cos alpha + cos beta)/(cos (alpha - beta))`

B

`(sin alpha - sin beta)/(sin (alpha - beta))`

C

`(cos alpha - cos beta)/(cos (alpha - beta))`

D

`(sin alpha + sin beta)/(sin(alpha + beta))`

Text Solution

Verified by Experts

The correct Answer is:
D
The equation fo a chord joining points having eccentric angles `alpha` and `beta` is given by
`x/a cos ((alpha + beta)/(2)) + y/b sin ((alpha + beta)/(2)) = cos ((alpha - beta)/(2))`
If the passes through (ae, 0) then
`e cos ((alpha + beta)/(2)) = cos ((alpha - beta)/(2))`
`rArr e = (cos((alpha - beta)/(2))/(cos ((alpha + beta)/(2))`
`rArr e = (2 sin ((alpha + beta)/(2))cos((alpha - beta)/(2))/2 sin ((alpha + beta)/(2))cos((alpha + beta)/(2)) = (sin alpha + sin beta)/(sin (alpha + beta))`
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