If $3 x+2 \sqrt{2} y+k=0$ is a normal to the hyperbola $4 x^2-9 y^2-36=0$ making positive intercepts on both the axes, then $k=$

If a hyperbola has asymptotes $3 x-4 y-1=0$ and $4 x-3 y-6=0$, then the transverse and conjugate axes of that hyperbola are

$x+y+3=0,2 x-y+1=0$ are the equations of the asymptotes of a hyperbola.

If $(1,-2)$ is a point on this hyperbola, then the equation of its conjugate hyperbola is

If $\theta$ is the acute angle between the tangents drawn from the point $(1,1)$ to the hyperbola $4 x^2-5 y^2-20=0$, then $\tan \theta=$