The value of m for which $y = mx + 6$ is a tangent to the hyperbola $\frac{{{x^2}}}{{100}} – \frac{{{y^2}}}{{49}} = 1$, is

The eccentricity of the hyperbola $\frac{{\sqrt {1999} }}{3}({x^2} – {y^2}) = 1$ is

The equation of the tangents to the conic $3{x^2} – {y^2} = 3$ perpendicular to the line $x + 3y = 2$ is

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $9 y^{2}-4 x^{2}=36$

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 $…….$.