91Ó°ÊÓ

If two conics have a common focus then show that two of their common chords pass through the point of intersection of their directrices.

If \(S\) be the focus, \(P\) and \(Q\) be two points on a conic such that the angle \(P S Q\) is constant, prove that the locus of the point of intersection of the tangents at \(P\) and \(O\) is a conic section whose focus is \(S\).

Prove that the two conics \(\frac{l_{1}}{r}=1+e_{1} \cos \theta\) and \(\frac{l_{2}}{r}=1+e \cos (\theta-\alpha)\) touch each other if \(l_{1}^{2}\left(1-e_{2}^{2}\right)+l_{2}^{2}\left(1-e_{1}^{2}\right)=2 l_{1} l_{2}\left(1-e_{1} e_{2} \cos \alpha\right)\).

Two equal ellipses of eccentricity \(e\) have one focus common and are placed with their axes at right angles. If \(P Q\) be a common tangent then prove that \(\sin \frac{1}{2} \angle P S Q=\frac{e}{\sqrt{2}}\)

A conic is described having the same focus and eccentricity as the conic \(\frac{l}{r}=1+e \cos \theta\) and the two conics touch at \(\theta=\alpha\). Prove that the length of its latus rectum is \(\frac{2 l\left(1-e^{2}\right)}{e^{2}+2 e \cos \alpha+1} .\)

If \(S M\) and \(S N\) be perpendiculars from the focus \(S\) on the tangent and normal at any point on the conic \(\frac{l}{r}=1+e \cos \theta\) and, \(S T\) the perpendicular on \(M N\) show that the locus to \(T\) is \(r\left(e^{2}-1\right)=e l \cos \theta\).

Prove that the exterior angle between any two tangents to a parabola is equal to half the difference of the vectorial angles of their points of contact.

Prove that if the chords of a conic subtend a constant angle at the focus, the tangents at the end of the chord will meet on a fixed conic and the chord will touch another fixed conic.