Show that the line touches the hyperbola x^2 a^2 - y^2 b^2 = 1 if a^2 ^2 - b^2 ^2 = p^2.

MATHEMATICS · Subjective · CBSE · Mathematics · Conic Sections · Conic Sections

Problem

Show that the line

x \cos \alpha + y \sin \alpha = p

touches the hyperbola

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad \text{if} \quad a^2 \cos^2 \alpha - b^2 \sin^2 \alpha = p^2.