Question

Find the minimum value of $|z-1|$ if $||z-3|-|z+1||=2$.

• A. $|z-1|\ge 0$
• B. $|z-1|\ge 1$
• C. $|z-1|\ge 2$
• D. $|z-1|\ge 3$

Answer 1

The correct option is B $|z-1|\ge 1$

We know that,

$|z_1+z_2|⩽||z_1|+|z_2||$

Therefore,

$2=||z-3|-|z+1||\le \left|\right(z-3)+(z+1\left)\right|$
$\Rightarrow |z-1|\ge 1$
Ans: B