If $(a\cos\alpha,\ b\sin{\alpha)}$, $(a\cos\beta,\ b\ \sin{\beta)}$ are the end points of a focal chord of an ellipse $b^{2}x^{2} + a^{2}y^{2} = a^{2}b^{2}$, then which of the following is correct?

Question

Dr. Abubacker · Accepted Answer

(b)
If $(a\cos\alpha,\ b\sin{\alpha)}$and $(a\cos\beta,\ b\sin{\beta)}$ are the end points of chord, then equation of chord is
$\frac{x}{a}\cos{\left( \frac{\alpha + \beta}{2} \right) + \frac{y}{b}\sin{\left( \frac{\alpha + \beta}{2} \right) = \cos\left( \frac{\alpha - \beta}{2} \right)}}$
If it is a focal chord, it passes through $(ae,\ 0)$, so
$e\cos\left( \frac{\alpha + \beta}{2} \right) = \cos\left( \frac{\alpha - \beta}{2} \right)$
$\Rightarrow e = \frac{\cos\left( \frac{\alpha - \beta}{2} \right)}{\cos\left( \frac{\alpha + \beta}{2} \right)}$

Q. If $(a\cos\alpha,\ b\sin{\alpha)}$, $(a\cos\beta,\ b\ \sin{\beta)}$ are the end points of a focal chord of an ellipse $b^{2}x^{2} + a^{2}y^{2} = a^{2}b^{2}$, then which of the following is correct?

Explanation

Reviewed by Dr. Abubacker

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