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 .............

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

If for a hyperbola the ratio of length of conjugate Axis to the length of transverse  axis is $3 : 2$ then the ratio of distance between the focii to the distance between the two directrices is

Let $P \left( x _0, y _0\right)$ be the point on the hyperbola $3 x ^2-4 y ^2$ $=36$, which is nearest to the line $3 x+2 y=1$. Then $\sqrt{2}\left( y _0- x _0\right)$ is equal to :

If $2 x-y+1=0$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{16}=1$, then which of the following $CANNOT$ be sides of a right angled triangle?

$[A]$ $2 a, 4,1$   $[B]$ $2 a, 8,1$   $[C]$ $a, 4,1$    $[D]$ $a, 4,2$

A normal to the hyperbola, $4x^2 - 9y^2\, = 36$ meets the co-ordinate axes $x$ and $y$ at $A$ and $B$, respectively . If the parallelogram $OABP$ ( $O$ being the origin) is formed, then the locus of $P$ is