How do you show that the tangents from the end points in a focal chord on a hyperbola meet at the directrix.
Equation of hyperbola: $ \dfrac {x^2} {a^2}- \dfrac {y^2} {b^2}=1 $
Original Question: Let $P (a\sec(\theta),b\tan(\theta))$ be a point on the hyperbola , with $\tan(\theta)$ not equal to $0$. The tangent at $P$ meets the directrix at $Q$ and the $S$ is the corresponding focus. $O$ is the origin. Prove that $SP$ is perpendicular to $SQ$.
Recall that in the pencil of rays from the focus $S $ the involution of conjugate lines respect to the conic is the elliptic involution of right angles. Thus the focus $S $ lies in the interior of the conic, hence the line $PS $ is secant and meets the conic at another point say $P'$. Then $Q $ is the pole of $PP'$. Consequently, $SP $ and $SQ $ are conjugate hence orthogonal.