The angle between the asymptotes of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is equal to:

If $y=mx+4$ ($n>0$ is a typo,likely $m>0$ or irrelevant) is a tangent to the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1$,then the point of contact of this tangent is

Let $H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1$ be the hyperbola,whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4\sqrt{3}$. Suppose the point $(\alpha, 6)$,where $\alpha > 0$,lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha, 6)$,then $\alpha^2+\beta$ is equal to:

If the eccentricities of the hyperbolas $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$ are $e$ and $e_1$ respectively,then $\frac{1}{e^2} + \frac{1}{e_1^2} = $

The eccentricity of the hyperbola whose length of the latus rectum is equal to $8$ and the length of its conjugate axis is equal to half of the distance between its foci is:

If the area of the quadrilateral formed by the tangents drawn at the ends of the latus rectum of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is equal to the square of the distance between the center and one focus of the hyperbola,then $e^3$ is ($e$ is the eccentricity of the hyperbola).