Properties of Hyperbola

If $S$ be the focus and $G$ be the point where the normal at $P$ meets the axis of hyperbola, then prove that $SG=eSP$ and the tangent and normal at $P$ bisects the external and internal angles between the focal distances of $P$.

If the sum of slopes of concurrent normals to the curve $xy=4$ is equal to the sum of ordinates of conormal points then locus of $P$ is

If the product of the slopes of the tangents drawn from an external point $P$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is a constant $k^2$, then the locus of $P$ is

If the hyperbola $x^2-\frac{y^2}{4}=1$ passes through the foci of the ellipse $\frac{x^2}{4}+\frac{y^2}{b^2}=1,(0<b<2)$ then the value of $b^2$ is

Four points are such that the line joining any two points is perpendicular to the line joining other two points. If three points out of these lie on a rectangular hyperbola, then the fourth point will lie on.

The locus of the foot of perpendicular drawn from the focus of the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1,$ to any arbitrary tangent of the hyperbola, is

If the ellipse $x^2+k^2{y}^2=k^2{a}^2$ is confocal with the hyperbola $x^2-y^2=a^2$, then -

The product of perpendicular drawn from any point on $\frac{x^2}{9}-\frac{y^2}{16}=1$ upon its asymptote is-

If (a sec $\theta$ , b tan $\theta$) and (a sec $\varphi$, b tan $\varphi$) are the ends of a focal chord of $\frac{{\text{x}}^2}{{\text{a}}^2}-\frac{{\text{y}}^2}{{\text{b}}^2}=1$, then $\text{tan}\frac{\theta }{2}\text{tan}\frac{\varphi }{2}$ is equal to

A ray emanating from the point $\left(5,0\right)$ is incident on the hyperbola $9x^2-16y^2=144$ at the point $P$ with abscissa $8$, then the equation of the reflected ray after first reflection is ($P$ lies in the first quadrant)

If ${\theta }_1 \& {\theta }_2$  are the parameters of the extremities of a chord through $\left(ae,0\right)$ of a hyperbola  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then $\tan \left(\frac{{\theta }_1}{2}\right)\tan \left(\frac{{\theta }_2}{2}\right)=$