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Maths-

General

Easy

Question

If $alpha$ and $beta$ are the eccentric angles of extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is -

$\frac{\cos α + \cos β}{\cos (α + β)}$
$\frac{\sin α - \sin β}{\sin (α - β)}$
$\frac{\cos α - \cos β}{\cos (α - β)}$
$\frac{\sin α + \sin β}{\sin (α + β)}$

The correct answer is: $fraction numerator sin invisible function application alpha plus sin invisible function application beta over denominator sin invisible function application left parenthesis alpha plus beta right parenthesis end fraction$

Equation of chord be
$fraction numerator x over denominator a end fraction$ cos $fraction numerator alpha plus beta over denominator 2 end fraction$ + $fraction numerator y over denominator b end fraction$ sin $fraction numerator alpha plus beta over denominator 2 end fraction$ = cos $fraction numerator alpha – beta over denominator 2 end fraction$
since it passes through (ae, 0)
so e cos $fraction numerator alpha plus beta over denominator 2 end fraction$ = cos $fraction numerator alpha – beta over denominator 2 end fraction$
e = $fraction numerator cos invisible function application fraction numerator alpha – beta over denominator 2 end fraction over denominator cos invisible function application fraction numerator alpha plus beta over denominator 2 end fraction end fraction$ = $fraction numerator 2 cos invisible function application fraction numerator alpha – beta over denominator 2 end fraction sin invisible function application fraction numerator alpha plus beta over denominator 2 end fraction over denominator sin invisible function application left parenthesis alpha plus beta right parenthesis end fraction$
e = $fraction numerator sin invisible function application alpha plus sin invisible function application beta over denominator sin invisible function application left parenthesis alpha plus beta right parenthesis end fraction$

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