What will be equation of that chord of hyperbola $25{x^2} - 16{y^2} = 400$, whose mid point is $(5, 3)$

Consider a branch of the hyperbola $x^2-2 y^2-2 \sqrt{2} x-4 \sqrt{2} y-6=0$ with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $\mathrm{C}$ is the focus of the hyperbola nearest to the point $\mathrm{A}$, then the area of the triangle $\mathrm{ABC}$ is

Let $S$ be the focus of the hyperbola $\frac{x^2}{3}-\frac{y^2}{5}=1$, on the positive $\mathrm{x}$-axis. Let $\mathrm{C}$ be the circle with its centre at $\mathrm{A}(\sqrt{6}, \sqrt{5})$ and passing through the point $\mathrm{S}$. if $\mathrm{O}$ is the origin and $\mathrm{SAB}$ is a diameter of $\mathrm{C}$ then the square of the area of the triangle $OSB$ is equal to ....................

The asymptote of the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}}= 1$ form with any tangent to the hyperbola a triangle whose area is $a^2$ $\tan$ $ \lambda $ in magnitude then its eccentricity is :

For hyperbola $\frac{{{x^2}}}{{{{\cos }^2}\alpha }} - \frac{{{y^2}}}{{{{\sin }^2}\alpha }} = 1$ which of the following remain constant if $\alpha$ varies

If the eccentricity of the standard hyperbola passing, through the point $(4, 6)$ is $2$, then the equation of the tangent to the hyperbola at $(4, 6)$ is