The equation of the normal at the point $(6, 4)$ on the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 3$ is:

If the line $y = 2x + \lambda$ is a tangent to the hyperbola $36x^2 - 25y^2 = 3600$,then $\lambda = $

For all real values of $m$,the straight line $y = mx + \sqrt{9m^2 - 4}$ is a tangent to the curve:

On a rectangular hyperbola $x^2-y^2=a^2, a > 0$,three points $A, B, C$ are taken as follows: $A=(-a, 0)$; $B$ and $C$ are placed symmetrically with respect to the $X$-axis on the branch of the hyperbola not containing $A$. Suppose that the $\triangle ABC$ is equilateral. If the side length of the $\triangle ABC$ is $ka$,then $k$ lies in the interval

The equation of the hyperbola whose foci are the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ and the eccentricity is $2$,is

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 $\sqrt{2}$)