The locus of the point of instruction of the lines $\sqrt 3 x - y - 4 \sqrt 3 t = 0$  $\&$  $\sqrt 3tx + ty - 4\sqrt 3 = 0$  (where $ t$  is a parameter) is a hyperbola whose eccentricity is

If the line $y\, = \,mx\, + \,7\sqrt 3 $ is normal to the hyperbola $\frac{{{x^2}}}{{24}} - \frac{{{y^2}}}{{18}} = 1,$ then a value of $m$ is 

The distance between the foci of a hyperbola is double the distance between its vertices and the length of its conjugate axis is $6$. The equation of the hyperbola referred to its axes as axes of co-ordinates is

Find the equation of the hyperbola with foci $(0,\,\pm 3)$ and vertices $(0,\,\pm \frac {\sqrt {11}}{2})$.

Find the equation of the hyperbola satisfying the give conditions : Vertices $(\pm 7,\,0)$,  $e=\frac{4}{3}$

An ellipse intersects the hyperbola $2 x^2-2 y^2=1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then

$(A)$ Equation of ellipse is $x^2+2 y^2=2$

$(B)$ The foci of ellipse are $( \pm 1,0)$

$(C)$ Equation of ellipse is $x^2+2 y^2=4$

$(D)$ The foci of ellipse are $( \pm \sqrt{2}, 0)$