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

(p) $\frac{1}{\sqrt{h^2+k^2}}=2 \Rightarrow h^2+k^2=\frac{1}{4}$ Hence, locus is a circle. (q) ||$z+2|-| z-2||=3$ and $\quad 2-(-2)=4>3$ Hence, locus is hyperbola. (r) Let $x=\sqrt{3}\left(\frac{1-t^2}{1+t^2}\right), y=\frac{2 t}{1+t^2}$ Let $\quad \tan \theta=t$ $\Rightarrow x=\sqrt{3} \cos 2 \theta, y=\sin 2 \theta$ $\therefore \quad \frac{x^2}{3}+y^2=1$ Hence, locus is an ellipse. (s) Eccentricity $x=1 \Rightarrow$ Parabola, $1 < x < \infty \Rightarrow$ Hyperbola (t) Let $z=x+i y$ Since, $\operatorname{Re}(z+1)^2=|z|^2+1$ $\Rightarrow \quad(x+1)^2-y^2=x^2+y^2+1$ $\Rightarrow \quad 2 x=2 y^2$ $\Rightarrow \quad x=y^2$ Hence, locus is parabola.