Question

In the conic $\frac{l}{r}=1-e\cos \theta$, find the envelope of chords which subtend a constant angle $2\alpha$ at the focus.

Answer 1

Suppose $AB$ is the chord of the given conic
$\frac{l}{r}1-e\cos \theta$ which subtends a constant angle $2\alpha$ at the focus of conic.

Let the vertival angle of A be $\beta -\alpha$; so the vectorial angle of $\beta$ is $\beta +\alpha$
Now, the equation to the chord AB will be
$\frac{l}{r}=\sec \alpha \cos (\theta -\beta )-e\cos \theta$
This always touches the conic
$\frac{l}{r}\cos \alpha =1-e\cos \alpha \cos \theta$