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

$A$ quadratic polynomial $y = f(x)$ with constant term $3$ neither touches nor intersects the $x$-axis and is symmetric about the line $x = 1$. The coefficient of the leading term of the polynomial is unity. $A$ point $A(x_1, y_1)$ with abscissa $x_1 = 1$ and a point $B(x_2, y_2)$ with ordinate $y_2 = 11$ are given in a Cartesian rectangular system of coordinates $OXY$ in the first quadrant on the curve $y = f(x)$,where $O$ is the origin. The vertex of the quadratic polynomial is:

Two mutually perpendicular tangents of the parabola $y^2 = 4ax$ meet the axis in $P_1$ and $P_2$. If $S$ is the focus of the parabola,then $\frac{1}{SP_1} + \frac{1}{SP_2}$ is equal to

The sum of the squares of the eccentricities of the conics $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ and $\frac{x^{2}}{4}-\frac{y^{2}}{3}=1$ is

Match the following parametric forms in List-$I$ with their corresponding conic sections in List-$II$:

List-$I$	List-$II$
$(A)$ $\left[\frac{p}{2}\left(t+\frac{1}{t}\right), \frac{q}{2}\left(t-\frac{1}{t}\right)\right]$	$(I)$ parabola
$(B)$ $(p+q \cos \theta, r+q \sin \theta)$	$(II)$ circle
$(C)$ $(p+\lambda^2, q-\lambda)$	$(III)$ ellipse
	$(IV)$ hyperbola

If the tangent to the parabola $y^2 = x$ at a point $(\alpha, \beta)$,$(\beta > 0)$ is also a tangent to the ellipse $x^2 + 2y^2 = 1$,then $\alpha$ is equal to