Chapter 9

If \(P S P^{\prime}\) and \(Q S Q^{\prime}\) are two focal chords of a conic cutting each other at right angles then prove that \(\frac{1}{S P . S P^{\prime}}+\frac{1}{S Q \cdot S Q^{\prime}}=\) a constant.

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

In a parabola with focus \(S\), the tangents at any points \(P\) and \(Q\) on it meet at \(T\). Prove that (i) \(S P \cdot S Q=S T^{2}\) (ii) The triangles \(S P T\) and \(S Q T\) are similar.

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 \(S\).

If the circle \(r+2 a \cos \theta=0\) cuts the conic \(\frac{1}{r}=1+e \cos (\theta-\alpha)\) in four real points find the equation in \(r\) which determines the distances of these points from the pole. Also, show that if their algebraic sum equals \(2 a\) and the eccentricity of the conic is \(2 \cos \alpha\).

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 \(l_{1}^{2}\left(1-e_{2}^{2}\right)+l_{2}^{2}\left(1-e_{1}^{2}\right)=2 l_{1} l_{2}\left(1-e_{1} e_{2} \cos \alpha\right)\).

Prove that the locus of the middle points of a system of focal chords of a conic section is a conic section which is a parabola, ellipse or hyperbola according as the original conic is a parabola ellipse or hyperbola.

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 \(\sin \frac{1}{2} \angle P S Q=\frac{e}{\sqrt{2}}\)

If the tangents at \(P\) and \(Q\) of a conic meet at a point \(T\) and \(S\) be the focus then prove that \(S T^{2}=S P \cdot S Q\) if the conic is a parabola.

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 \(\frac{2 l\left(1-e^{2}\right)}{e^{2}+2 e \cos \alpha+1} .\)

Prove that three normals can be drawn from a given point to a given parabola. If the normal at \(\alpha, \beta, \gamma\) on the conic \(\frac{l}{r}=1+\cos \theta\) meet at the point \((\rho, \phi)\) prove that \(\phi=\frac{\alpha+\beta+\gamma}{2}\).

If the normals at three points of the parabola \(r=a \cos e c^{2} \frac{\theta}{2}\) whose vectorial angles are \(\alpha, \beta, \gamma\) meet in a point whose vectorial angle is \(\phi\) then prove that \(2 \phi=\alpha+\beta+\gamma-\pi\)

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 \(r\left(e^{2}-1\right)=e l \cos \theta\).

If the tangent at any point of an ellipse make an angle \(\alpha\) with its major axis and an angle \(\beta\) with the focal radius to the point of contact then show that \(e \cos \alpha=\cos \beta . \quad\) Ans.: \(A^{2}+B^{2}-2 e(A \cos \gamma+B \sin \gamma)+e^{2}-1=0\)

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.

If the tangent and normal at any point \(P\) of a conic meet the transverse axis of \(T\) and \(G\), respectively, and if \(S\) be the focus then prove that \(\frac{1}{S G}-\frac{1}{S T}\) is a constant.