The equation of the pair of asymptotes of the hyperbola $x^2-2 y^2-8 x+8 y+4=0$ is

The product of the perpendicular distances from any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ to its asymptotes is

For the hyperbola $x^2-y^2-4x+2y+c=0$,if the focus is $S(2+2\sqrt{2}, k)$ and the directrix that is adjacent to $S$ is $x=2+\sqrt{2}$,then $c=$

If one of the roots of the equation $x^2 - 5x - 14 = 0$ is the length of the semi-conjugate axis of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and the square of the other root is the semi-transverse axis,then the focus of the hyperbola that lies on the positive $x$-axis is

The equation of the director circle of the hyperbola $\frac{x^2}{16} - \frac{y^2}{4} = 1$ is given by

Let $P$ be a point on the hyperbola $x^2 - y^2 = 4$,which is at the minimum distance from $(0, -1)$. Then the distance of $P$ from the $x$-axis is: