The curve $xy = c, (c > 0)$, and the circle $x^2 + y^2 = 1$ touch at two points. Then the distance between the points of contacts is
$1$
$2$
$2\sqrt 2 $
None of these
The product of the lengths of perpendiculars drawn from any point on the hyperbola $x^2 -2y^2 -2=0$ to its asymptotes is
If the tangent and normal to a rectangular hyperbola $xy = c^2$ at a variable point cut off intercept $a_1, a_2$ on $x-$ axis and $b_1, b_2$ on $y-$ axis, then $(a_1a_2 + b_1b_2)$ is
The locus of the point of intersection of any two perpendicular tangents to the hyperbola is a circle which is called the director circle of the hyperbola, then the eqn of this circle is
The equation of the hyperbola whose foci are the foci of the ellipse $\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{9} = 1$ and the eccentricity is $2$, is
The line $3x - 4y = 5$ is a tangent to the hyperbola ${x^2} - 4{y^2} = 5$. The point of contact is