What is the slope of the tangent line drawn to the hyperbola $xy = a\,(a \ne 0)$ at the point $(a, 1)$

The value of $m$, for which the line $y = mx + \frac{{25\sqrt 3 }}{3}$, is a normal to the conic $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1$, is

Let the focal chord of the parabola $P: y^{2}=4 x$ along the line $L: y=m x+c, m>0$ meet the parabola at the points $M$ and $N$. Let the line $L$ be a tangent to the hyperbola $H : x ^{2}- y ^{2}=4$. If $O$ is the vertex of $P$ and $F$ is the focus of $H$ on the positive $x$-axis, then the area of the quadrilateral $OMFN$ is.

The equation to the chord joining two points $(x_1, y_1)$  and $(x_2, y_2)$  on the rectangular hyperbola $xy = c^2$  is

The equation to the hyperbola having its eccentricity $2$ and the distance between its foci is $8$

The eccentricity of the hyperbola can never be equal to