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Maths-

General

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Question

If $alpha comma beta$ are the eccentric angles of the 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: $fraction numerator sin invisible function application alpha plus sin invisible function application beta over denominator sin invisible function application left parenthesis alpha plus beta right parenthesis end fraction$

$open parentheses a cos invisible function application alpha comma b sin invisible function application alpha close parentheses comma open parentheses a cos invisible function application beta comma b sin invisible function application beta close parentheses comma open parentheses a e comma 0 close parentheses$ are collinear.
$fraction numerator b left parenthesis sin invisible function application beta – sin invisible function application alpha right parenthesis over denominator a left parenthesis cos invisible function application beta – cos invisible function application alpha right parenthesis end fraction equals fraction numerator b sin invisible function application alpha – 0 over denominator a cos invisible function application alpha – a e end fraction$
$therefore left parenthesis c o s invisible function application alpha minus e right parenthesis left parenthesis s i n invisible function application beta minus s i n invisible function application alpha right parenthesis equals s i n invisible function application alpha left parenthesis c o s invisible function application beta minus c o s invisible function application alpha right parenthesis$
$therefore$ e = $blank fraction numerator cos invisible function application alpha left parenthesis sin invisible function application beta – sin invisible function application alpha right parenthesis – sin invisible function application alpha left parenthesis cos invisible function application beta – cos invisible function application alpha right parenthesis over denominator sin invisible function application beta – sin invisible function application alpha end fraction$ = $fraction numerator sin invisible function application left parenthesis alpha – beta right parenthesis over denominator sin invisible function application alpha – sin invisible function application beta end fraction blank$ = $fraction numerator sin invisible function application alpha plus sin invisible function application beta over denominator sin invisible function application left parenthesis alpha plus beta right parenthesis end fraction$

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