Question

If $z$ and $w$ be two complex numbers such that $|z|\le 1,|w|\le 1$ and $|z+iw|=|z-i\overline{w}|=2,$then

• A. $|z|=|w|=\frac{1}{2}$
• B. $|z|=\frac{1}{2},|w|=\frac{3}{4}$
• C. $|z|=|w|=\frac{3}{4}$
• D. $|z|=|w|=1$

Answer 1

The correct option is D $|z|=|w|=1$

$\because |z+iw|\le |z|+|iw|$
$\Rightarrow 2\le |z|+|i||w|$
$\Rightarrow |z|+|w|\ge 2$
But from given condition, $|z|+|w|\le 2$
$\therefore |z|=|w|=1$