Consider a hyperbola $\mathrm{H}$ having centre at the origin and foci and the $\mathrm{x}$-axis. Let $\mathrm{C}_1$ be the circle touching the hyperbola $\mathrm{H}$ and having the centre at the origin. Let $\mathrm{C}_2$ be the circle touching the hyperbola $\mathrm{H}$ at its vertex and having the centre at one of its foci. If areas (in sq. units) of $\mathrm{C}_1$ and $\mathrm{C}_2$ are $36 \pi$ and $4 \pi$, respectively, then the length (in units) of latus rectum of $\mathrm{H}$ is

The locus of middle points of the chords of the circle $x^2 + y^2 = a^2$ which touch the hyperbola $\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1$ is

The locus of the point of intersection of the lines $(\sqrt{3}) kx + ky -4 \sqrt{3}=0$ and $\sqrt{3} x-y-4(\sqrt{3}) k=0$ is a conic, whose eccentricity is ………….

Let $a>0, b>0$. Let $e$ and $\ell$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1$. Let $e ^{\prime}$ and $\ell^{\prime}$ respectively the eccentricity and length of the latus rectum of its conjugate hyperbola. If $e ^{2}=\frac{11}{14} \ell$ and $\left( e ^{\prime}\right)^{2}=\frac{11}{8} \ell^{\prime}$, then the value of $77 a+44 b$ is equal to

 Find the equation of the hyperbola where foci are $(0,\,±12)$ and the length of the latus rectum is $36.$