Question

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

Answer 1

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

On squaring and subtracting, we get: 

\[ x^2 - y^2 = 64 \sec^2 \theta - 64 \tan^2 \theta\]

\[ \Rightarrow x^2 - y^2 = 64\]

\[ \Rightarrow \frac{x^2}{64} - \frac{y^2}{64} = 1\]

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

\[\Rightarrow \frac{2 \times 64}{\sqrt{64 + 64}}\]

\[ = \frac{128}{8\sqrt{2}}\]

\[ = \frac{16}{\sqrt{2}}\]

\[ = 8\sqrt{2}\]