Mathematics - Hyperbola Question with Solution | TestHub

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