<s>[INST] Proving that a given line is tangent to a hyperbola The question is: a line $x \cos\theta + y\sin\theta = p$ is given such that $a^2\cos^2\theta - b^2\sin^2\theta =p^2$.
I have to prove that it touches a hyperbola  $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. I do not see how to proceed. I thought I could proceed by eliminating $p$ by squaring the line equation and equate both but I don't get an idea what to do next. What do I need to show to prove that it touches that hyperbola?
 [/INST] The tangent at $(x_1,y_1)$ to the hyperbola is 
\begin{align*}
\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1
\end{align*}
If this is the given line, we have
\begin{align*}
\frac{x_1/a^2}{\cos \theta} = \frac{-y_1/b^2}{\sin \theta} = \frac{1}{p}
\end{align*}
Hence $x_1 = \frac{a^2\cos\theta}{p}, y_1 = -\frac{b^2\sin\theta}{p}$. Since this lies on the curve, we have
\begin{align*}
a^2\cos^2\theta - b^2\sin^2\theta = p^2
\end{align*}
</s>