If the lengths of the transverse and conjugate axes of a hyperbola are $8$ and $6$ respectively,find the difference of the distances of any point on the hyperbola from its foci.

The difference of the focal distances of any point on the hyperbola $9x^2 - 16y^2 = 144$ is

The line $21x + 5y = k$ touches the hyperbola $7x^2 - 5y^2 = 232$. Then $k =$

Let $A(2 \sec \theta, 3 \tan \theta)$ and $B(2 \sec \phi, 3 \tan \phi)$ where $\theta+\phi=\frac{\pi}{2}$ be two points on the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$. If $(\alpha, \beta)$ is the point of intersection of normals to the hyperbola at $A$ and $B$,then $\beta$ is equal to

If the variable line $y = kx + 2h$ is tangent to an ellipse $2x^2 + 3y^2 = 6$,the locus of $P(h, k)$ is a conic $C$ whose eccentricity equals

If a hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $(\pm 2, 0)$,then the tangent to this hyperbola at $P$ is: