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

Column $$I$$
(A) Circle
(B) Parabola
(C) Ellipse
(D) Hyperbola

Column $$II$$
(p) The locus of the point $$(h, k)$$ for which the line $$hx+ky=1$$ touches the circle $${x^2} + {y^2} = 4$$
(q) Points $$z$$ in the complex plane satisfying $$\left| {z + 2} \right| - \left| {z - 2} \right| = \pm 3$$
(r) Points of the conic have parametric representation $$x = \sqrt 3 \left( {{{1 - {t^2}} \over {1 + {t^2}}}} \right),\,\,y = {{2t} \over {1 + {t^2}}}$$
(s) The eccentricity of the conic lies in the interval $$1 \le x \le \infty $$
(t) Points $$z$$ in the complex plane satisfying $${\mathop{\rm Re}\nolimits} \,{\left( {z + 1} \right)^2}\, = {\left| z \right|^2} + 1$$

Match the statements in Column $$I$$ with the properties in Column $$II$$ and indicate your answer by darkening the appropriate bubbles in the $$4 \times 4$$ matrix given in the $$ORS$$.

Column $$I$$
(A) Two intersecting circles
(B) Two mutually external circles
(C) Two circles, one strictly inside the other
(D) Two branches vof a hyperbola

Column $$II$$
(p) have a common tangent
(q) have a common normal
(r) do not have a common tangent
(s) do not have a common normal

Find the equation of the common tangent in $${1^{st}}$$ quadrant to the circle $${x^2} + {y^2} = 16$$ and the ellipse $${{{x^2}} \over {25}} + {{{y^2}} \over 4} = 1$$. Also find the length of the intercept of the tangent between the coordinate axes.

Tangents are drawn from any point on the hyperbola $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$ to the circle $${x^2} + {y^2} = 9$$.Find the locus of mid-point of the chord of contact.