Question

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

• A. $\frac{7\sqrt{7}}{2\sqrt{3}}$
• B. $\frac{5\sqrt{7}}{2\sqrt{3}}$
• C. $\frac{14\sqrt{7}}{\sqrt{3}}$
• D. $\frac{7\sqrt{7}}{5\sqrt{3}}$

Answer 1

The correct option is B $\frac{5\sqrt{7}}{2\sqrt{3}}$

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