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:

Find the area of the region enclosed by the equation $2|x| + 3|y| = 6$ in square units.

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:

Let the foci of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{7}=1$ and the hyperbola $\frac{x^{2}}{144}-\frac{y^{2}}{\alpha}=\frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola is:

If the line $y = \sqrt{3}x$ cuts the curve $x^4 + ax^2y + bxy + cx + dy + 6 = 0$ at $A$,$B$,$C$,and $D$,then the value of $OA \cdot OB \cdot OC \cdot OD$ is,(where $O$ is the origin).

Let the hyperbola $H : \frac{x^2}{a^2} - y^2 = 1$ and the ellipse $E : 3x^2 + 4y^2 = 12$ be such that the length of the latus rectum of $H$ is equal to the length of the latus rectum of $E$. If $e_H$ and $e_E$ are the eccentricities of $H$ and $E$ respectively,then the value of $12(e_H^2 + e_E^2)$ is equal to