The distance between the tangent lines to the hyperbola $x^2-2y^2=18$ which are perpendicular to the line $y=x$ is

Let $P(a \sec \theta, b \tan \theta)$ and $Q(a \sec \phi, b \tan \phi)$ be two points on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,where $\theta + \phi = \frac{\pi}{2}$. If $(h, k)$ is the point of intersection of the normals at $P$ and $Q$,then $k = \dots$

If the line $x \cos \alpha + y \sin \alpha = p$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,then:

Let a line $L_{1}$ be tangent to the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{4}=1$ and let $L_{2}$ be the line passing through the origin and perpendicular to $L_{1}$. If the locus of the point of intersection of $L_{1}$ and $L_{2}$ is $(x^{2}+y^{2})^{2} = \alpha x^{2}+\beta y^{2}$,then $\alpha+\beta$ is equal to

Let $LL^{\prime}$ be the latus rectum through the focus $S$ of a hyperbola and $A^{\prime}$ be the opposite vertex of the hyperbola. If triangle $A^{\prime}LL^{\prime}$ is equilateral,then the eccentricity of the hyperbola is

Tangent at any point $\theta$ on the curve $x=35 \sec \theta, y=35 \tan \theta$ is