The square of the slope of a common tangent drawn to the circle $4x^2 + 4y^2 = 25$ and the ellipse $4x^2 + 9y^2 = 36$ is

The graph of the conic $x^2-(y-1)^2=1$ has one tangent line with positive slope that passes through the origin. If the point of tangency is $(a, b)$,then find the value of $\sin^{-1}\left(\frac{a}{b}\right)$.

If $S \equiv \frac{x^2}{k-7}+\frac{y^2}{11-k}-1=0, k \in R-\{7,11\}$,then which one of the following statements is incorrect?

Match the conics in Column-$I$ with the statements/expressions in Column-$II$.

Column-$I$	Column-$II$
$A$. Circle	$P$. Locus of point $(h, k)$ such that the line $hx + ky = 1$ touches the circle $x^2 + y^2 = 4$
$B$. Parabola	$Q$. Point $z$ in the complex plane satisfies $|z + 2| - |z - 2| = \pm 3$
$C$. Hyperbola	$R$. Eccentricity of the conic lies in the interval $[1, \infty)$
	$S$. Point $z$ in the complex plane satisfies $Re(z + 1)^2 = |z|^2 + 1$

The tangent and normal at $P(t),$ for all real positive $t,$ to the parabola $y^2 = 4ax$ meet the axis of the parabola in $T$ and $G$ respectively. Find the angle at which the tangent at $P$ to the parabola is inclined to the tangent at $P$ to the circle passing through the points $P, T,$ and $G$.

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