The point of contact of the line $y = x - 1$ with the hyperbola $3x^2 - 4y^2 = 12$ is

If the equation of the hyperbola with foci $(4,2)$ and $(8,2)$ is $3x^2-y^2-\alpha x+\beta y+\gamma=0$,then $\alpha+\beta+\gamma$ is equal to . . . . . . .

Find the equation of the hyperbola whose foci are $(6, 5)$ and $(-4, 5)$ and eccentricity is $5/4$.

If the center,vertex,and focus of a hyperbola are $(0, 0)$,$(4, 0)$,and $(6, 0)$ respectively,then the equation of the hyperbola is:

The difference between the focal distances of any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $6$. If $(\sqrt{13}, k)$ is an endpoint of a latus rectum of this hyperbola,then $k=$

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