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:

The equation $x^2 - 4y^2 - 2x + 16y - 40 = 0$ represents:

Let $S = \{(x,y) \in \mathbb{R}^2 : \frac{y^2}{1+r} - \frac{x^2}{1-r} = 1\}$,where $r \neq \pm 1$. Then $S$ represents

If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with centre $C$ are such that $CP$ is perpendicular to $CQ$,where $a < b$,then the value of $\frac{1}{(CP)^2} + \frac{1}{(CQ)^2}$ is:

If $(1,2)$ is the focus,$x+2y=0$ is the directrix,and $\sqrt{2}$ is the eccentricity of a hyperbola,then the equation of the hyperbola is

If the foci of a hyperbola are the same as that of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and the eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse,then the smaller focal distance of the point $\left(\sqrt{2}, \frac{14}{3} \sqrt{\frac{2}{5}}\right)$ on the hyperbola is equal to