Let $H$ be the hyperbola,whose foci are $(1 \pm \sqrt{2}, 0)$ and eccentricity is $e = \sqrt{2}$. Then the length of its latus rectum is

The distance between the directrices of a rectangular hyperbola is $10$ units. What is the distance between its foci?

Find the equation of the hyperbola satisfying the given conditions: Vertices $(0, \pm 3)$,foci $(0, \pm 5)$.

The locus of the middle points of the chords of the hyperbola $3x^2 - 2y^2 + 4x - 6y = 0$ parallel to $y = 2x$ is

If $3 \sqrt{2} x - 4 y = 12$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $\frac{5}{4}$ is its eccentricity,then $a^2 - b^2 =$

If a circle cuts a rectangular hyperbola $xy = c^2$ at four points $A, B, C,$ and $D$,and the parameters of these four points are $t_1, t_2, t_3,$ and $t_4$ respectively,then which of the following is true?