If $\mathrm{e}_{1}$ and $\mathrm{e}_{2}$ are the eccentricities of the ellipse, $\frac{\mathrm{x}^{2}}{18}+\frac{\mathrm{y}^{2}}{4}=1$ and the hyperbola, $\frac{\mathrm{x}^{2}}{9}-\frac{\mathrm{y}^{2}}{4}=1$ respectively and $\left(\mathrm{e}_{1}, \mathrm{e}_{2}\right)$ is a point on the ellipse, $15 \mathrm{x}^{2}+3 \mathrm{y}^{2}=\mathrm{k},$ then $\mathrm{k}$ is equal to 

The equation of the hyperbola whose foci are $(-2, 0)$ and $(2, 0)$ and eccentricity is $2$ is given by :-

Let the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ be $\frac{5}{4}$. If the equation of the normal at the point $\left(\frac{8}{\sqrt{5}}, \frac{12}{5}\right)$ on the hyperbola is $8 \sqrt{5} x +\beta y =\lambda$, then $\lambda-\beta$ is equal to

The locus of the point of intersection of the lines $ax\sec \theta + by\tan \theta = a$ and $ax\tan \theta + by\sec \theta = b$, where $\theta $ is the parameter, is

The product of the lengths of perpendiculars drawn from any point on the hyperbola ${x^2} - 2{y^2} - 2 = 0$ to its asymptotes is

With one focus of the hyperbola $\frac{{{x^2}}}{9}\,\, - \,\,\frac{{{y^2}}}{{16}}\,\, = \,\,1$ as the centre , a circle is drawn which is tangent to the hyperbola with no part of the circle being outside the hyperbola. The radius of the circle is