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I'm currently reading through a document about the ellipse. I've attached the provided image and working out.

enter image description here

From here, it is easy enough to show that $|OP|\sin\gamma=|FP|\sin\alpha$ using say the Sine Rule.

However, they follow through with

enter image description here

I understand how they got the first two lines using the polar equation of the ellipse (with respect to the focus) $r=FP=\frac{a(1-e^2)}{1+e\cos\theta}$.

Where I have issue is with the "Summing the squares". I cannot see how they managed to get rid of all the $\alpha$ and have everything in terms of $\gamma$.

Is there some sort of identity that I am missing?

Trogdor
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1 Answers1

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Divide the first equation by $\sqrt{1-e^2}$: $$ \begin{align} |OP|\frac{\sin{\gamma}}{\sqrt{1-e^2}}&=|OQ|\frac{\sqrt{1-e^2}\sin(\alpha)}{1+e\cos(\alpha)}\tag{1}\\ |OP|\cos(\gamma)&=|OQ|\frac{e+\cos(\alpha)}{1+e\cos(\alpha)}\tag{2} \end{align} $$ then add the squares of $(1)$ and $(2)$: $$ |OP|^2\left(\frac{\sin^2(\gamma)}{1-e^2}+\cos^2(\gamma)\right)=|OQ|^2\tag{3} $$

robjohn
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