• # question_answer The minimum value of $|{{z}_{1}}-{{z}_{2}}|$ as ${{z}_{1}}$ and ${{z}_{2}}$ vary over the curve  $|\sqrt{3}(1-2z)+2i|=2\sqrt{7}$and $|\sqrt{3}(-1-z)-2i|=\sqrt{3}(9-z)+18i|$ respectively, is [Note: $i=\sqrt{-1}$] A)  $\frac{7\sqrt{7}}{2\sqrt{3}}$                   B)  $\frac{5\sqrt{7}}{2\sqrt{3}}$ C)  $\frac{14\sqrt{7}}{\sqrt{3}}$                   D)  $\frac{7\sqrt{7}}{5\sqrt{3}}$

$|\,\sqrt{3}\,(1-2z)+2i\,|\,=2\sqrt{7}$ is the equation of circle having centre is $\left( \frac{1}{2},\frac{1}{\sqrt{3}} \right)$ and having radius$\frac{\sqrt{7}}{3}$. Also, $\left| \,\sqrt{3}(1-2z)-2i \right|=\left| \sqrt{3}(9-z)+18i \right|$ Is the equation of perpendicular bisector of line joining $\left( -1,\frac{-2}{\sqrt{3}} \right)$ and $\left( 9,6\sqrt{3} \right)$ So, $MQ=\sqrt{{{\left( 4-\frac{1}{2} \right)}^{2}}+{{\left( \frac{8}{\sqrt{3}}-\frac{1}{\sqrt{3}} \right)}^{2}}}=\frac{7\sqrt{7}}{2\sqrt{3}}$ $\therefore$Required distance $=(MQ)-$ (radius) $=\frac{7\sqrt{7}}{2\sqrt{3}}-\sqrt{\frac{7}{3}}=\frac{5}{2}\sqrt{\frac{7}{3}}$