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If alphaa n dbeta are the eccentric a...

If `alphaa n dbeta` 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"(alphabeta+))`

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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Knowledge Check

  • If alpha, beta are the eccentric angles of the extermities of a focal chord of an ellipse, then eccentricity of the ellipse is

    A
    `(sin alpha+ sin beta)/(sin(alpha+beta))`
    B
    `(cos alpha+ cos beta)/(cos(alpha+beta))`
    C
    `(sin(alpha+beta))/(sin(alpha+beta))`
    D
    none of these

  • Find the eccentricity of the ellipse (2)/( sqrt3)

    A
    `(2)/( sqrt3)`
    B
    `(sqrt3)/(2)`
    C
    `(1)/( sqrt3)`
    D
    `(3)/( sqrt2)`

  • If alpha,beta are the eccentric angles of the extremities of a focal chord of the ellipse x^(2)/16 + y^(2)/9 = 1 , then tan (alpha/2) tan(beta/2) =

    A
    `(sqrt7+4)/(sqrt7-4)`
    B
    `-9/23`
    C
    `(sqrt5-4)/(sqrt5+4)`
    D
    `(8sqrt7-23)/9`

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