The equation of the tangents to the conic $3{x^2} - {y^2} = 3$ perpendicular to the line $x + 3y = 2$ is
$y = 3x \pm \sqrt 6 $
$y = 6x \pm \sqrt 3 $
$y = x \pm \sqrt 6 $
$y = 3x \pm 6$
The combined equation of the asymptotes of the hyperbola $2{x^2} + 5xy + 2{y^2} + 4x + 5y = 0$
Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is reciprocal to that of the hyperbola $2 x^2-2 y^2=1$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is $................$.
The equation of a common tangent to the conics $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ and $\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1$ is
The equation of the normal to the hyperbola $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1$ at $( - 4,\;0)$ is
The equation of the normal at the point $(a\sec \theta ,\;b\tan \theta )$ of the curve ${b^2}{x^2} - {a^2}{y^2} = {a^2}{b^2}$ is