Hyperbola and Rectangular Hyperbola — NDA Mathematics PYQs

What is the distance between the two foci of the hyperbola $25x^2-75y^2=225$?

What is the distance between the foci of the hyperbola $x^2-4y^2=1$?

Consider the following statements in respect of hyperbola $\frac{x^2}{{\cos }^2\theta }-\frac{y^2}{{\sin }^2\theta }=1$:

1. The two foci are independent of $\theta$.
2. The eccentricity is $\sec \theta$.
3. The distance between the two foci is $2$ units.
   How many of the statements given above are correct?

If any point on a hyperbola is $(3\tan \theta ,2\sec \theta )$, then what is the eccentricity of the hyperbola?

The equation $2x^2-3y^2-6=0$ represents

What is the equation of the hyperbola having latus rectum and eccentricity 8 and $\frac{3}{\sqrt{5}}$ respectively?

What is the eccentricity of rectangular hyperbola?