The locus of the mid points of the chords of the hyperbola $\mathrm{x}^{2}-\mathrm{y}^{2}=4$, which touch the parabola $\mathrm{y}^{2}=8 \mathrm{x}$, is :

At the point of intersection of the rectangular hyperbola $ xy = c^2 $ and the parabola $y^2 = 4ax$  tangents to the rectangular hyperbola and the parabola make an angle $ \theta $ and $ \phi $ respectively with the axis of $X$, then

The equation of the hyperbola referred to the axis as axes of co-ordinate and whose distance between the foci is $16$ and eccentricity is $\sqrt 2 $, is

Let $H _{ n }=\frac{ x ^2}{1+ n }-\frac{ y ^2}{3+ n }=1, n \in N$. Let $k$ be the smallest even value of $n$ such that the eccentricity of $H _{ k }$ is a rational number. If $l$ is length of the latus return of $H _{ k }$, then $21 l$ is equal to $.......$.

Let a line $L: 2 x+y=k, k\,>\,0$ be a tangent to the hyperbola $x^{2}-y^{2}=3 .$ If $L$ is also a tangent to the parabola $y^{2}=\alpha x$, then $\alpha$ is equal to :

If the line $x-1=0$, is a directrix of the hyperbola $kx ^{2}- y ^{2}=6$, then the hyperbola passes through the point.