If $ PN$  is the perpendicular from a point on a rectangular hyperbola $x^2 - y^2 = a^2 $ on any of its asymptotes, then the locus of the mid point of $PN$  is :

The equation of the tangent to the hyperbola $2{x^2} - 3{y^2} = 6$ which is parallel to the line $y = 3x + 4$, is

The tangent at an extremity (in the first quadrant) of latus rectum of the hyperbola $\frac{{{x^2}}}{4} - \frac{{{y^2}}}{5} = 1$ , meet $x-$ axis and $y-$ axis at $A$ and $B$ respectively. Then $(OA)^2 - (OB)^2$ , where $O$ is the origin, equals

$P$  is a point on the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}}$ $= 1, N $ is the foot of the perpendicular from $P$  on the transverse axis. The tangent to the hyperbola at $P$  meets the transverse axis at $ T$  . If $O$ is the centre of the hyperbola, the $OT. ON$  is equal to :

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 $.......$.

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