Let $P \left( x _0, y _0\right)$ be the point on the hyperbola $3 x ^2-4 y ^2$ $=36$, which is nearest to the line $3 x+2 y=1$. Then $\sqrt{2}\left( y _0- x _0\right)$ is equal to :

If for a hyperbola the ratio of length of conjugate Axis to the length of transverse  axis is $3 : 2$ then the ratio of distance between the focii to the distance between the two directrices is

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 :

Tangents are drawn to the hyperbola $4{x^2} - {y^2} = 36$ at the points $P$ and $Q.$ If these tangents intersect at the point $T(0,3)$ then the area (in sq. units) of $\Delta PTQ$ is :

The eccentricity of the hyperbola $5{x^2} - 4{y^2} + 20x + 8y = 4$ is

$C$ the centre of the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$. The tangents at any point $P$ on this hyperbola meets the straight lines $bx - ay = 0$ and $bx + ay = 0$ in the points $Q$ and $R$ respectively. Then $CQ\;.\;CR = $