If a circle of radius $4 \text{ cm}$ passes through the foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{4} = 1$ and is concentric with the hyperbola,then the eccentricity of the conjugate hyperbola of that hyperbola is

What is the eccentricity of the conjugate hyperbola of the hyperbola $x^2 - 3y^2 = 1$?

Let $S$ denote the locus of the point of intersection of the pair of lines $4x - 3y = 12\alpha$ and $4\alpha x + 3\alpha y = 12$, where $\alpha$ varies over the set of non-zero real numbers. Let $T$ be the tangent to $S$ passing through the points $(p, 0)$ and $(0, q)$, with $q > 0$, and parallel to the line $4x - \frac{3}{\sqrt{2}}y = 0$. Then the value of $pq$ is (in $\sqrt{2}$)

If $(\alpha, -1)$ is an interior point of the curve $4x^2 - 3y^2 = 1$,then $\alpha$ lies in:

If a hyperbola has asymptotes $3x - 4y - 1 = 0$ and $4x - 3y - 6 = 0$,then the transverse and conjugate axes of that hyperbola are

Let $P(3,3)$ be a point on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. If the normal to it at $P$ intersects the $x$-axis at $(9,0)$ and $e$ is its eccentricity,then the ordered pair $(a^{2}, e^{2})$ is equal to