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

If the curves $\frac{x^2}{a^2} + \frac{y^2}{4} = 1$ and $y^3 = 16x$ intersect at right angles,then $a^2 =$

Which one of the following is the common tangent to the ellipses $\frac{x^2}{a^2 + b^2} + \frac{y^2}{b^2} = 1$ and $\frac{x^2}{a^2} + \frac{y^2}{a^2 + b^2} = 1$?

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 $e_{1}^{2}+e_{2}^{2}$.

Consider the circle $x^2+y^2=9$ and the parabola $y^2=8x$. They intersect at $P$ and $Q$ in the first and the fourth quadrants,respectively. Tangents to the circle at $P$ and $Q$ intersect the $x$-axis at $R$ and tangents to the parabola at $P$ and $Q$ intersect the $x$-axis at $S$.
$1.$ The ratio of the areas of the triangles $PQS$ and $PQR$ is
$(A)$ $1:\sqrt{2}$ $(B)$ $1:2$ $(C)$ $1:4$ $(D)$ $1:8$
$2.$ The radius of the circumcircle of the triangle $PRS$ is
$(A)$ $5$ $(B)$ $3\sqrt{3}$ $(C)$ $3\sqrt{2}$ $(D)$ $2\sqrt{3}$
$3.$ The radius of the incircle of the triangle $PQR$ is
$(A)$ $4$ $(B)$ $3$ $(C)$ $8/3$ $(D)$ $2$
Give the answer for questions $1, 2$ and $3.$

Find the equation of the ellipse whose focus is $(-1, 1)$,eccentricity is $1/2$,and directrix is $x - y + 3 = 0$.