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.

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Solution

As the conics have the same focus and the same directrix, their equations may be given by,
$\frac{a e_{1}}{r} = 1 - e_{1} cos θ$ ....(1)
$\frac{a e_{2}}{r} = 1 - e_{2} cos θ$ ....(2)
The equation to the tangent at point $^{'} α^{'}$ of $1$ will be
$\frac{a e_{1}}{r} = cos (θ - α) - e_{1} cos θ$ ....()
Let it meet (2) at the point $ϕ$ , then we have
$\frac{a e_{1}}{r} = cos (ϕ - α) - e_{1} cos ϕ$ ....(4)
Dividing $4$ by $2$ , we get
$\frac{e_{1}}{e_{2}} = \frac{cos (ϕ - α) - e_{1} cos ϕ}{1 - e_{2} cos ϕ}$
$\Rightarrow e_{1} - e_{1} e_{2} cos ϕ = e_{2} cos (ϕ - α) - e_{1} e_{2} cos ϕ$
$\Rightarrow cos (ϕ - α) = \frac{e_{1}}{e_{2}}$ , which is a constant quantity
Hence, $ϕ - α$ must be constant.

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