The line $3x - 4y = 5$ is a tangent to the hyperbola $x^2 - 4y^2 = 5$. The point of contact is

Find the coordinates of the foci and the vertices,the eccentricity,and the length of the latus rectum of the hyperbola $49 y^{2}-16 x^{2}=784$.

$A$ point on the curve $\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ is

Find the equation of the hyperbola satisfying the given conditions: Foci $(0, \pm 13)$,the conjugate axis is of length $24$.

The asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,with any tangent to the hyperbola form a triangle whose area is $a^2 \tan (\alpha)$. Then its eccentricity equals

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