The eccentricity of a hyperbola passing through the points $(3, 0)$ and $(3\sqrt{2}, 2)$ is:

For the hyperbola $x^2 \sec^2 \theta - y^2 \csc^2 \theta = 1$,which of the following remains constant when $\theta$ varies?

Find the equation of the hyperbola with foci $(0, \pm 3)$ and vertices $(0, \pm \frac{\sqrt{11}}{2})$.

The tangent to the hyperbola $xy = c^2$ at the point $P(ct, c/t)$ intersects the $x$-axis at $T$ and the $y$-axis at $T'$. The normal to the hyperbola at $P$ intersects the $x$-axis at $N$ and the $y$-axis at $N'$. If the areas of the triangles $PNT$ and $PN'T'$ are $\Delta$ and $\Delta'$ respectively,then $\frac{1}{\Delta} + \frac{1}{\Delta'}$ is:

If $l$ is the maximum value of $-3x^2+4x+1$ and $m$ is the minimum value of $3x^2+4x+1$,then the equation of the hyperbola having foci at $(l, 0)$ and $(7m, 0)$ and eccentricity $e=2$ is

The intersection of two perpendicular tangents to the hyperbola $\frac{x^2}{4} - \frac{y^2}{2} = 1$ lies on the circle $x^2 + y^2 = \dots \dots \dots$