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Corresponding Points : Ellipse

If α and ...

Question

If $α$ and $β$ are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is:

A

\frac{cos α + cos β}{cos (α - β)}

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B

\frac{sin α - sin β}{sin (α - β)}

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C

\frac{cos α - cos β}{cos (α - β)}

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D

\frac{sin α + sin β}{sin (α + β)}

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Solution

The correct option is B $\frac{sin α + sin β}{sin (α + β)}$
The equation of a chord joining points having eccentric angles $α$ and $β$ is given by

$\frac{x}{a} c o s (\frac{α + β}{2}) + \frac{y}{b} s i n (\frac{α + β}{2}) = c o s (\frac{α - β}{2})$

If it passes through $(a e, 0)$ then

$e c o s (\frac{α + β}{2}) = c o s (\frac{α - β}{2})$

$\Rightarrow e = \frac{c o s (\frac{α - β}{2})}{c o s (\frac{α + β}{2})}$

$\Rightarrow \frac{2 s i n (\frac{α + β}{2}) c o s (\frac{α - β}{2})}{2 s i n (\frac{α + β}{2}) c o s (\frac{α + β}{2})} = \frac{s i n α + s i n β}{s i n (α + β)}$

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