The equation of a common tangent to the curves $y^2 = 16x$ and $xy = -4$ is:

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 $3.$

If $e_1$,$e_2$,and $e_3$ are eccentricities of the conics $y = x^2 - x + 3$,$\frac{x^2}{a^2} + \frac{y^2}{3a^4} = 1$,and $a^2x^2 - 3a^4y^2 = 1$ respectively,then which of the following is correct? (where $a > 1$)

The locus of the foot of the perpendicular from the centre of the hyperbola $xy = c^2$ on a variable tangent is :

An ellipse has eccentricity $\frac{1}{2}$ and one focus at the point $P\left( \frac{1}{2}, 1 \right)$. Its one directrix is the common tangent nearer to the point $P$, to the circle $x^2 + y^2 = 1$ and the hyperbola $x^2 - y^2 = 1$. The equation of the ellipse in the standard form is:

If the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1$ coincide,then $b^2$ is equal to