If the line $y = mx + c$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,then the locus of the point $P(m, c)$ is:

Find the equation of the hyperbola satisfying the given conditions: Vertices $(0, \pm 3)$,foci $(0, \pm 5)$.

At which of the following points is the tangent to the hyperbola $x^2 - y^2 = 3$ parallel to the line $2x + y + 8 = 0$?

The equation of a line passing through the centre of a rectangular hyperbola is $x - y - 1 = 0$. If one of the asymptotes is $3x - 4y - 6 = 0$,the equation of the other asymptote is:

The equation of the hyperbola whose coordinates of the foci are $(\pm 8, 0)$ and the length of the latus rectum is $24$ units, is

The locus of the centroid of the triangle formed by any point $P$ on the hyperbola $16x^{2}-9y^{2}+32x+36y-164=0$ and its foci is: