If $e_{1}$ is the eccentricity of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $a > b$,and $e_{2}$ is the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$,then find the value of $e_{1}^{2}+e_{2}^{2}$.

The equation of the common tangent to the curves ${y^2} = 8x$ and $xy = -1$ is

The circle $x^2 + y^2 = 5$ meets the parabola $y^2 = 4x$ at $P$ and $Q$. Then the length $PQ$ is equal to:

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 locus of the foot of the perpendicular from the centre of the hyperbola $xy = c^2$ on a variable tangent is :

The circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ intersect at points $A$ and $B$. The equation of the common tangent with a positive slope to the circle and the hyperbola is: