$A$ 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

Let $P$ be a point on the hyperbola $x^2 - y^2 = 4$,which is at the minimum distance from $(0, -1)$. Then the distance of $P$ from the $x$-axis is:

Let $H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be a hyperbola such that the distance between its foci is $6$ and the distance between its directrices is $\frac{8}{3}$. If the line $x = k$ intersects the hyperbola $H$ at the points $A$ and $B$ such that the area of the triangle $AOB$ is $4\sqrt{15}$,where $O$ is the origin,then $a^2$ equals

If the distance between the foci and the distance between the directrices of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ are in the ratio $3: 2$,then $a: b$ is

If the equation of the tangent drawn at $(h, k)$ to the hyperbola $\frac{(x-1)^2}{1}-\frac{(y-2)^2}{2}=1$ is $x=2$,then $h+k=$

The length of the latus rectum of the hyperbola $\frac{x^2}{\cos^2 \alpha} - \frac{y^2}{\sin^2 \alpha} = 4$ is (where $\alpha \neq \frac{n\pi}{2}, n \in I$).