Question

Write the distance between the directrices of the hyperbola x = 8 sec θ, y = 8 tan θ.

Answer 1

We have:
$x=8\sec \theta ,y=8\tan \theta$

$$\begin{gathered} \mathrm{On}\mathrm{squaring}\mathrm{and}\mathrm{subtracting},\mathrm{we}\mathrm{get}: \\ {x}^2-y^2=64{\sec }^2\theta -64{\tan }^2\theta \\ \Rightarrow x^2-y^2=64 \\ \Rightarrow \frac{x^2}{64}-\frac{y^2}{64}=1 \end{gathered}$$

∴ a = b = 8
Distance between the directrices of hyperbola is $\frac{2a^2}{\sqrt{a^2+b^2}}$.

$$\begin{gathered} \Rightarrow \frac{2\times 64}{\sqrt{64+64}} \\ =\frac{128}{8\sqrt{2}} \\ =\frac{16}{\sqrt{2}} \\ =8\sqrt{2} \end{gathered}$$