Locus of intersecting perpendicular tangents of conic, using polar coordinates

Question

Find the locus of point of intersection of two tangents to the conic $$\frac{l}{r}=1+e \cos\theta$$ which are at right angles to one another.

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My approach: It's straight forward for Cartesian coordinates. But in polar coordinates on writing equation of tangent lines in polar form I got stuck in elimination. Little new to polar coordinates for such kind of problems. Any suggestions are welcome.

Thanks!

Answer 1

I have done for ellipse, you can do the same for hyperbola.

Knowing that the locus is a circle (called director circle) centered at the center of the ellipse, the equation of the circle in polar coordinates centered at $(r_0, \phi)$ and radius $R$ is given as $$r=r_0\cos(\theta-\phi)\pm\sqrt{R^2-r_0^2\sin^2(\theta-\phi)}$$

For the ellipse $\displaystyle\frac{l}{r}=1+e\cos\theta$, we have center $(-ae,π)$ and radius $a^2-b^2$, given that $\displaystyle l=\frac{b^2}{a}$.

$$r=-ae\cos\theta\pm\sqrt{2b^2+a^2e^2\cos^2\theta}$$

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This was the most efficient method to find the locus i.e. first find the locus in Cartesian coordinate system and then convert it into polar one.

If you wish to find the locus directly in polar coordinates, it will be quite (or heavily) tedious. However, if you still want to proceed, here's a start: equation of tangent to the given conic at a point with polar angle $\alpha$ will be $$\frac{l}{r}=e\cos\theta+\cos(\theta-\alpha).$$