The equation of the hyperbola whose directrix is $x + 2y = 1$, focus $(2, 1)$ and eccentricity $2$ will be
${x^2} - 16xy - 11{y^2} - 12x + 6y + 21 = 0$
$3{x^2} + 16xy + 15{y^2} - 4x - 14y - 1 = 0$
${x^2} + 16xy + 11{y^2} - 12x - 6y + 21 = 0$
None of these
The condition that the straight line $lx + my = n$ may be a normal to the hyperbola ${b^2}{x^2} - {a^2}{y^2} = {a^2}{b^2}$ is given by
A normal to the hyperbola, $4x^2 - 9y^2\, = 36$ meets the co-ordinate axes $x$ and $y$ at $A$ and $B$, respectively . If the parallelogram $OABP$ ( $O$ being the origin) is formed, then the locus of $P$ is
The one which does not represent a hyperbola is
The equation of the normal at the point $(6, 4)$ on the hyperbola $\frac{{{x^2}}}{9} - \frac{{{y^2}}}{{16}} = 3$, is
If $e$ and $e’$ are eccentricities of hyperbola and its conjugate respectively, then