Hyperbola

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.

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 $m$.

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

The latus rectum of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $8$ and length of conjugate axis is equal to half the distance between the foci, then its eccentricity is

If the equation of the hyperbola whose foci at $\left(0,\pm 6\right)$ and length of conjugate axis$=2\sqrt{11}$ is of the form $m{y}^2-n{x}^2=275$ then find the value of $m+n$.

If the equation of the hyperbola whose vertices are at $\left(0,\pm 6\right)$ and eccentricity $=\frac{5}{3}$ is of the form $m{y}^2-n{x}^2=576$ then find the value of $m+n$.

The equation of the hyperbola whose foci are $\left(0,\pm 12\right)$ and length of the latus rectum$=36$ is

The equation of the hyperbola whose vertices are $\left(0,\pm 3\right)$ and foci are $\left(0,\pm 5\right)$ is

If the coordinates of foci of the hyperbola $\frac{y^2}{9}-\frac{x^2}{27}=1$ are of the form $\left(0,\pm k\right)$ then find the value of $k$.

If the eccentricity of the hyperbola $y^2-x^2=1$ is $\sqrt{k}$ then find the value of $k$.

Find the sum of the lengths of transverse axis and conjugate axis of the hyperbola $y^2-16x^2=16$.

The equation of the hyperbola whose vertices are $\left(0,\pm 5\right)$ and foci are $\left(0,\pm 8\right)$ is

If the coordinates of the foci of the hyperbola $16x^2-9y^2=144$ are of the form $\left(\pm k,0\right)$ then find the value of $k$.

If $\left(3,4\right)$ is a point on the hyperbola having foci $\left(3,0\right)$ and $\left(\lambda ,0\right)$, if the length of the transverse axis being $1$ unit, then find the values of $\lambda$.