The equation of a common tangent to the parabola $y^2 = 4x$ and the hyperbola $xy = 2$ is

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.

What is the area of the quadrilateral formed by the line $|x| + |y| = 1$?

The locus of the middle point of the intercept of the tangents drawn to the ellipse $x^2 + 2y^2 = 2$ between the coordinate axes is:

The angle between the curves $x^2-y^2=4$ and $x^2+y^2=4\sqrt{2}$ is

Let $e_1$ and $e_2$ be two distinct roots of the equation $x^2 - ax + 2 = 0$.  Let the sets  $S_1 = \{a \in \mathbb{R} : e_1 \text{ and } e_2 \text{ are the eccentricities of hyperbolas} \} = (\alpha, \beta),$ and $S_2 = \{a \in \mathbb{R} : e_1 \text{ and } e_2 \text{ are the eccentricities of an ellipse and a hyperbola, respectively} \} = (\gamma, \infty).$ Then $\alpha^2 + \beta^2 + \gamma^2$ is equal to: