Equation of a Hyperbola

Let $A\left({\theta }_1\right)$ and $B\left({\theta }_2\right)$ be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $S$ be the focus of the hyperbola. If $A,S,B$ are collinear and $a\cos \left(\frac{{\theta }_1+{\theta }_2}{2}\right)=k\cos \left(\frac{{\theta }_1-{\theta }_2}{2}\right)$ then $k=$

If $\frac{x^2}{k-\frac{5}{2}}+\frac{y^2}{\frac{7}{3}-k}=1$ ($k$ is a real number) represents a hyperbola, then the set of all values of $k$ is

If $P\left(\theta \right)=\left(x_1,\frac{3\sqrt{5}}{2}\right),0<\theta <\frac{\pi }{2}$ is a point on the hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1$, where $\theta$ is the parameter in its parametric form, then $2x_1+9{\sin }^2\theta =$

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

If the transverse and conjugate axes of hyperbola are equal, then its eccentricity 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

If the eccentricities of the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$ and the hyperbola $\frac{x^2}{64}-\frac{y^2}{b^2}=1$ are reciprocals of each other, then $b^2$ is equal to