The auxiliary circle of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is given by:

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

The locus of the point of intersection of lines $(x + y)t = a$ and $x - y = at$,where $t$ is the parameter,is

The foci of the hyperbola $\frac{x^2}{16} - \frac{(y - 2)^2}{9} = 1$ are:

$A$ normal to the hyperbola $4x^2 - 9y^2 = 36$ meets the coordinate axes $x$ and $y$ at $A$ and $B$,respectively. If the parallelogram $OABP$ ($O$ being the origin) is formed,then the locus of $P$ is

Find the eccentricity of a hyperbola whose latus rectum is $8$ and whose conjugate axis is half the distance between its foci.