The asymptote of the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}}= 1$ form with any tangent to the hyperbola a triangle whose area is $a^2$ $\tan$ $ \lambda $ in magnitude then its eccentricity is :

Let the latus rectum of the hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ subtend an angle of $\frac{\pi}{3}$ at the centre of the hyperbola. If $\mathrm{b}^2$ is equal to $\frac{l}{\mathrm{~m}}(1+\sqrt{\mathrm{n}})$, where $l$ and $\mathrm{m}$ are co-prime numbers, then $l^2+\mathrm{m}^2+\mathrm{n}^2$ is equal to______________.

The line $lx + my + n = 0$ will be a tangent to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$, if

If the two tangents drawn on hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ in such a way that the product of their gradients is ${c^2}$, then they intersects on the curve

The distance between the directrices of a rectangular hyperbola is $10$ units, then distance between its foci is

Let tangents drawn from point $C(0,-b)$ to hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ touches hyperbola at points $A$ and $B.$ If $\Delta ABC$ is a right angled triangle, then $\frac{a^2}{b^2}$ is equal to -