Basics of Hyperbola

Prove that the square of the length of a focal chord of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ having slope $m$ is $\frac{4a^2{b}^4{\left(1+m^2\right)}^2}{{\left(a^2{m}^2-b^2\right)}^2}$.

The equation $y^2-x^2=2$ represents a rectangular hyperbola.

The equation $y^2-x^2=64$ represents a rectangular hyperbola.

The equation $y^2-x^2=16$ represents a rectangular hyperbola.

The equation $y^2-x^2=4$ represents a rectangular hyperbola.

The equation $y^2-x^2=1$ represents a rectangular hyperbola.

The equation $y^2-x^2+2x-1=0$ represents a rectangular hyperbola.

The sound of a cannon firing is heard one second later at a position $B$ than at position $A$. If the speed of sound is uniform, then

Find the length of the latus-rectum of the rectangular hyperbola $x^2-y^2-81=0$.

Find the length of the latus-rectum of the rectangular hyperbola $x^2-y^2-9=0$.

Find the length of the latus-rectum of the rectangular hyperbola $x^2-y^2-25=0$.

 What type of conic is represented by the equation $x^2-y^2=9?$  What is its eccentricity?

Show that the eccentricity of any rectangular hyperbola is $\sqrt{2}$.

The equation $y^2-x^2+2x-1=0$ represents a hyperbola.

Let $\theta \in \left(0,\frac{\pi }{2}\right).$ If the eccentricity of the hyperbola $x^2{\cos }^2\theta -y^2=6{\cos }^2\theta$ is $\sqrt{3}$ times the eccentricity of the ellipse $x^2+y^2{\cos }^2\theta =30{\cos }^2\theta ,$ then $\theta$ is equal to

Find the equation of the hyperbola whose foci are $\left(4,1\right)$ and $\left(8,1\right)$ and whose eccentricity is $2$.

The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes through the point of intersection of the lines $7x+13y-87=0 \mathrm{and} 5x-8y+7=0$ and its latus rectum is of length $\frac{32\sqrt{2}}{5}$. Eccentricity is $\frac{\sqrt{m}}{5}$. Find $m$.

Sum of the lengths of transverse axis and conjugate axis of the following hyperbola $y^2-16x^2=16$ is

The length of the latus-rectum of the following hyperbola $9y^2-4x^2=72$ is

The difference in focal distances of any point on the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ is