Hyperbola

The difference between the distance of any point on the hyperbola from the two foci is $\mathbf{1 6}$ and the eccentricity is $\mathbf{2}$. Then the equation of the hyperbola is

The foci of a hyperbola are the same as those of the ellipse with equation $9 x^2+16 y^2=144$.
If the length of the transverse axis of this hyperbola is $2 \cos \alpha$, then its equation is:

If the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the foci of the hyperbola $\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}$ coincide, then the value of $b^2$ is

The length of the latus rectum of a conic $49 y^2-16 x^2=784$ is

If the distance between the foci and the distance between the two directrixes are in the ratio \(3: 2\) for a hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\), then a : b is

The distance between the foci of a hyperbola is 16 and its eccentricity is \(\sqrt{2}\). Then its equation is

If \({{{x^2}} \over {36}} - {{{y^2}} \over {{k^2}}} = 1\) is a hyperbola, then which of the following statements can be true?