$AB$ is a double ordinate of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ such that $\Delta AOB$ (where $O$ is the origin) is an equilateral triangle. Then the eccentricity $e$ of the hyperbola satisfies:

Let the focal chord of the parabola $P: y^{2}=4x$ along the line $L: y=mx+c, m>0$ meet the parabola at the points $M$ and $N$. Let the line $L$ be a tangent to the hyperbola $H: x^{2}-y^{2}=4$. If $O$ is the vertex of $P$ and $F$ is the focus of $H$ on the positive $x$-axis,then the area of the quadrilateral $OMFN$ is.

Let the circle $C$ touch the line $x - y + 1 = 0$,have the centre on the positive $x$-axis,and cut off a chord of length $\frac{4}{\sqrt{13}}$ along the line $-3x + 2y = 1$. Let $H$ be the hyperbola $\frac{x^2}{\alpha^2} - \frac{y^2}{\beta^2} = 1$,whose one of the foci is the centre of $C$ and the length of the transverse axis is the diameter of $C$. Then $2\alpha^2 + 3\beta^2$ is equal to . . . . . .

If the curves $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and $\frac{x^2}{l^2} - \frac{y^2}{m^2} = 1$ cut each other orthogonally,then :-

Match the conics in Column-$I$ with the statements/expressions in Column-$II$.

Column-$I$	Column-$II$
$A$. Circle	$P$. Locus of point $(h, k)$ such that the line $hx + ky = 1$ touches the circle $x^2 + y^2 = 4$
$B$. Parabola	$Q$. Point $z$ in the complex plane satisfies $|z + 2| - |z - 2| = \pm 3$
$C$. Hyperbola	$R$. Eccentricity of the conic lies in the interval $[1, \infty)$
	$S$. Point $z$ in the complex plane satisfies $Re(z + 1)^2 = |z|^2 + 1$

Find the angle of intersection of the curves $y^{2}=4ax$ and $x^{2}=4by$.