Question

Two conic sections have the same focus and directrix. Show that any tangent from the outer curve to the inner one subtends a constant angle at the focus.

Answer 1

As the conics have the same focus and the same directrix, their equations may be given by,
$\frac{a{e}_1}{r}=1-e_1\cos \theta$ ....(1)
$\frac{a{e}_2}{r}=1-e_2\cos \theta$ ....(2)

The equation to the tangent at point ${}^{\prime }{\alpha }^{\prime }$ of $1$ will be

$\frac{a{e}_1}{r}=\cos (\theta -\alpha )-e_1\cos \theta$ ....()
Let it meet (2) at the point $\varphi$, then we have
$\frac{a{e}_1}{r}=\cos (\varphi -\alpha )-e_1\cos \varphi$ ....(4)
Dividing $4$ by $2$, we get
$\frac{e_1}{e_2}=\frac{\cos (\varphi -\alpha )-e_1\cos \varphi }{1-e_2\cos \varphi }$
$\Rightarrow e_1-e_1{e}_2\cos \varphi =e_2\cos (\varphi -\alpha )-e_1{e}_2\cos \varphi$
$\Rightarrow \cos (\varphi -\alpha )=\frac{e_1}{e_2}$, which is a constant quantity
Hence, $\varphi -\alpha$ must be constant.