How do we find locus of P looking at two foci distance $a$ apart such that
$$ \cos \phi/\cos \theta =- k~? $$
(k positive real). Vaguely remember it comes from Electromagnetic theory. Thanks in advance.
$$ \frac{r}{\sqrt{r^2+a^2-2 ar \cos \theta}}=\frac{\sin \phi}{\sin \theta}= \frac{\sqrt{1-k^2 \cos^2 \theta}}{\sin \theta}$$
Not in explicit polar form.
Starting with $$x=r\cos\theta, y=r\sin\theta$$ Note that $$\cos\theta=\frac{x}{\sqrt{x^2+y^2}}$$ and that $$\sin^2\phi=1-k^2\cos^2\theta$$
The Sine Rule gives $$\frac{\sin(\phi-\theta)}{a}=\frac{\sin\phi}{r}$$ $$\implies r\sin\phi\cos\theta-r\cos\phi\sin\theta=a\sin\phi$$ $$\implies (x-a)\sin\phi=y\cos\phi$$
Hence, $$(x-a)^2\left(1-k^2\frac{x^2}{x^2+y^2}\right)=y^2k^2\frac{x^2}{x^2+y^2}$$
So the Cartesian equation of the locus of $P$ is $$(x-a)^2\left(x^2(1-k^2)+y^2\right)=k^2x^2y^2$$