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:

The angle between the curves $2x^2 + y^2 = 20$ and $4y^2 - x^2 = 8$ at a point where they intersect in the $4^{th}$ quadrant is

If the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1$ coincide,then $b^2$ is equal to

The equation of common tangents to the parabola $y^2 = 8x$ and the hyperbola $3x^2 - y^2 = 3$ is:

If a normal drawn to the ellipse $\frac{x^2}{4} + \frac{y^2}{3} = 1$ touches the hyperbola $\frac{x^2}{4} - \frac{y^2}{3} = 1$,then the square of the slope of that normal is

Suppose that the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ are $(f_1, 0)$ and $(f_2, 0)$ where $f_1 > 0$ and $f_2 < 0$. Let $P_1$ and $P_2$ be two parabolas with a common vertex at $(0,0)$ and with foci at $(f_1, 0)$ and $(2f_2, 0)$,respectively. Let $T_1$ be a tangent to $P_1$ which passes through $(2f_2, 0)$ and $T_2$ be a tangent to $P_2$ which passes through $(f_1, 0)$. If $m_1$ is the slope of $T_1$ and $m_2$ is the slope of $T_2$,then the value of $(\frac{1}{m_1^2} + m_2^2)$ is