With one focus of the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$ as the centre,a circle is drawn which is tangent to the hyperbola with no part of the circle being outside the hyperbola. The radius of the circle is

For some $\theta \in \left(0, \frac{\pi}{2}\right),$ if the eccentricity of the hyperbola $x^{2} - y^{2} \sec^{2} \theta = 10$ is $\sqrt{5}$ times the eccentricity of the ellipse $x^{2} \sec^{2} \theta + y^{2} = 5,$ then the length of the latus rectum of the ellipse is

The ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $(b>a)$ and the parabola $y^2=4ax$ intersect at right angles. If $e$ is the eccentricity of the ellipse,then $2e^2=$

Through the vertex $O$ of the parabola $y^2 = 4ax$,two chords $OP$ and $OQ$ are drawn,and the circles on $OP$ and $OQ$ as diameters intersect in $R$. If $\theta_1, \theta_2$,and $\phi$ are the angles made with the axis by the tangents at $P$ and $Q$ on the parabola and by $OR$ respectively,then the value of $\cot \theta_1 + \cot \theta_2$ is:

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 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 $\sin^{-1}\left(\frac{a}{b}\right)$.