Let S be the focus of the hyperbola $xy=1$. Let a tangent to the hyperbola at point P cuts the latus rectum (through S) produced, at point Q and the directrix (corresponding to S) at point T. Also let M be the foot of perpendicular drawn from the point P to the same directrix. If angle PTS=$\theta_1$ and angle PMS=$\theta_2$, find $\frac{\theta_1}{\theta_2}$ and $\frac{SQ}{ST}$
My Attempt:
I made the diagram and guess that P,M,T,S lies on a circle. Not sure though.
Tangent at P is $\frac{x}{x_1}+\frac{y}{y_1}=2$, where $(x_1,y_1)$ are the coordinates of P.
Taking x-axis as the directrix, PM=$y_1$, MT=$√2-x_1$
Taking S as $(√2,√2)$
Not able to proceed ahead.
Best Answer
We have $\angle PMT=90°$ by construction and $\angle PST=90°$ by a well-known property of any conic section. Hence quadrilateral $PSTM$ is cyclic and $\angle PMS=\angle PTS$.
Line $QS$ is perpendicular by construction to $PM$, while $PS\perp ST$ as seen above. Hence $\angle QST=\angle SPM$ because the sides of these angles are pairwise perpendicular. It follows that triangles $QST$ and $SPM$ are similar and $SQ/ST=PS/PM=e=\sqrt2$.