Question

If $z, \bar{z},-z,-\bar{z}$ forms a rectangle of area $2 \sqrt{3}$ square units, then one such $z$ is

• A. $\frac{1}{2}+\sqrt{3} i$
• B. $\frac{\sqrt{5}+\sqrt{3} i}{4}$
• C. $\frac{3}{2}+\frac{\sqrt{3} i}{2}$
• D. $\frac{\sqrt{3}+\sqrt{11} i}{2}$

Answer 1

Answer: (A)

Let $z=x+i y$
Then, vertices of rectangle are $(x, y),(x,-y),(-x,-y),(-x, y)$
Now, area of rectangle $=(2 x)(2 y)=4 x y$

It is given that,
$\therefore 4 x y=2 \sqrt{3} $
$\Rightarrow 2 x y=\sqrt{3}$
$x=\frac{1}{2}, y=\sqrt{3} $
$\therefore z=\frac{1}{2}+\sqrt{3} i$