If two tangents are drawn to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ such that the product of their gradients is $c^2$,then they intersect on the curve:

If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with centre $C$ are such that $CP$ is perpendicular to $CQ$,where $a < b$,then the value of $\frac{1}{(CP)^2} + \frac{1}{(CQ)^2}$ is:

Find the equation of the hyperbola satisfying the given conditions: Foci $(\pm 4, 0)$,the latus rectum is of length $12$.

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

At which point does the line $2x + \sqrt{6}y = 2$ touch the hyperbola $x^2 - 2y^2 = 4$?

If the eccentricity of a hyperbola is $\frac{5}{3}$,then the eccentricity of its conjugate hyperbola is