Equation of Hyperbola in Standard Form-Parametric Equations

Show that the angle between the two asymptotes of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $2{\tan }^{-1}\left(\frac{b}{a}\right)$ or $2{\sec }^{-1}\left(e\right)$.

Prove that the product of the perpendicular distances from any point on a hyperbola to its asymptotes is constant. 

Find the equation of the hyperbola of given length of transverse axis $6$ whose vertex bisects the distance between the center and the focus.

Find the equation to the hyperbola whose foci are $(4,2)\text{ and }(8,2)$ and eccentricity is $2.$

Find the center, foci, eccentricity, equation of directrices, length of latus rectum of the hyperbola $9x^2-16y^2+72x-32y-16=0$.

Find the center, foci, eccentricity, equation of directrices, length of latus rectum of the hyperbola $5x^2-4y^2+20x+8y=4$.

Find the center, foci, eccentricity, equation of directrices, length of latus rectum of the hyperbola $x^2-4y^2=4$.

Find the center, foci, eccentricity, equation of directrices, length of latus rectum of the hyperbola $16y^2-9x^2=144$.

If the angle between asymptotes is $30°$ then find its eccentricity.

Find the equation of the hyperbola whose asymptotes are $3x=\pm 5y$ and the vertices are $(\pm 5,0)$.

If the eccentricity of a hyperbola is $\frac{5}{4}$, then find the eccentricity of its conjugate hyperbola.

Find the product (in fraction form) of lengths of the perpendiculars from any point on the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ to its asymptotes.

Find the equation of the hyperbola, whose asymptotes are the straight lines $(x+2y+3)=0,(3x+4y+5)=0$ and which passes through the point $(1,-1)$.

Find the equation of the hyperbola whose foci are $(\pm 5,0)$, the transverse axis is of length $8$.

If the lines $3x-4y=12\text{ and }3x+4y=12$ meets on a hyperbola $S=0$ then find the eccentricity of the hyperbola $S=0$.

One focus of a hyperbola is located at the point $(1,-3)$ and the corresponding directrix is the line $y=2$. Find the equation of the hyperbola if its eccentricity is $\frac{3}{2}$.