The distance between the tangents drawn to the hyperbola $3x^2 - y^2 = 3$,which are parallel to the line $y = 2x + 4$,is:

Consider a hyperbola $H : x^{2}-2y^{2}=4$. Let the tangent at a point $P(4, \sqrt{6})$ meet the $x$-axis at $Q$ and the latus rectum at $R(x_{1}, y_{1})$,where $x_{1}>0$. If $F$ is a focus of $H$ which is nearer to the point $P$,then the area of $\Delta QFR$ is equal to ....... .

If $e_1$ and $e_2$ are respectively the eccentricities of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and its conjugate hyperbola,then the line $\frac{x}{2 e_1}+\frac{y}{2 e_2}=1$ touches the circle having centre at the origin. Find its radius.

If the line $y = 2x + \lambda$ is a tangent to the hyperbola $36x^2 - 25y^2 = 3600$, then $\lambda = $

What is the area of the triangle formed by the lines $x - y = 0$,$x + y = 0$,and any tangent to the hyperbola $x^{2} - y^{2} = a^{2}$?

The asymptotes 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 $e$ is: