Equation of a Hyperbola

If the vertices and foci of a hyperbola are respectively $\left(\pm 3,0\right)$ and $\left(\pm 4,0\right)$ then the parametric equations of that hyperbola are

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.

If the latus rectum subtends a right angle at center of the hyperbola, then its eccentricity is:

The equation of hyperbola whose eccentricity is $\frac{5}{3}$ and distance between the foci is $10$ units is

In a hyperbola, if the length of transverse axis is twice that of the conjugate axis, then the distance between its directrices is $\_\_\_$ units.

If the transverse and conjugate axes of hyperbola are equal, then its eccentricity is $\_\_\_\_\_\_\_\_\_\_\_\_$

If in a hyperbola, the distance between the foci is $10$ units and transverse axis has length $8$ units, then the length of its latus rectum (in units) is:

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 $2$.

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 eccentricity of the hyperbola conjugate to $x^2-3y^2=2x+8$ is

The eccentricity of the hyperbola whose latus-rectum is half of its transverse axis, is

If $\frac{x^2}{36}-\frac{y^2}{k^2}=1$ is a hyperbola, then which of the following point lies on hyperbola?

Equation of the hyperbola whose vertices are $(\pm 3, 0)$ and foci at $(\pm 5, 0)$ is