The line $3x - 4y = 5$ is a tangent to the hyperbola ${x^2} - 4{y^2} = 5$. The point of contact is

If the product of the perpendicular distances from any point on the hyperbola $\frac{{{x^2}}}{{{a^2}}}\,\, - \,\,\frac{{{y^2}}}{{{b^2}}}\,\,\, = \,1$ of eccentricity $e =\sqrt 3 \,$ from its asymptotes is equal to $6$, then the length of the transverse axis of the hyperbola is

If $\frac{{{{\left( {3x - 4y - z} \right)}^2}}}{{100}} - {\frac{{\left( {4x + 3y - 1} \right)}}{{225}}^2} = 1$ then
length of latusrectum of hyperbola is

With one focus of the hyperbola $\frac{{{x^2}}}{9}\,\, - \,\,\frac{{{y^2}}}{{16}}\,\, = \,\,1$ as the centre , a circle is drawn which is tangent to the hyperbola with no part of the circle being outside the hyperbola. The radius of the circle is

The equation of the normal at the point $(6, 4)$ on the hyperbola $\frac{{{x^2}}}{9} - \frac{{{y^2}}}{{16}} = 3$, is

Latus rectum of the conic satisfying the differential equation, $ x dy + y dx = 0$  and passing through the point $ (2, 8) $ is :