If the line $4x + 3y = 7$ touches the hyperbola $x^2 - y^2 = 7$, then the sum of the coordinates of the point of contact 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

The distance between the tangent lines to the hyperbola $x^2-2y^2=18$ which are perpendicular to the line $y=x$ is

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

Find the coordinates of the foci and the vertices,the eccentricity,and the length of the latus rectum of the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$.

The locus of the point of intersection of the lines $(\sqrt{3})kx + ky - 4\sqrt{3} = 0$ and $\sqrt{3}x - y - 4\sqrt{3}k = 0$ is a conic,whose eccentricity is .............