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

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

Let $P, Q, R, S$ be the points of intersection of the circle $x^2+y^2=4$ and the hyperbola $xy=\sqrt{3}$. If $P=(\alpha, \beta)$ and $\alpha>\beta>0$,then the equation of the tangent drawn at $P$ to the hyperbola is

The condition that the straight line $lx + my = n$ may be a normal to the hyperbola $b^2x^2 - a^2y^2 = a^2b^2$ is given by

If $c$ is a real number and $\frac{x^2}{c-12}+\frac{y^2}{7-c}=1$ represents a hyperbola,then

The equation of the pair of asymptotes of the hyperbola $4x^2 - 9y^2 - 24x - 36y - 36 = 0$ is