Question

If z is any complex number such that $|3z-2|+|3z+2|=4$.

Then prove that the locus of z is a line segment.

Answer 1

$|3z-2|+|3z+2|=4$
$⟹|z-\frac{2}{3}|+|z+\frac{2}{3}|=\frac{4}{3}$ (1)
If $P\left(z\right)$ be any point $A=(2/3,0)$, $B=(-2/3,0)$, then (1) represents $PA+PB=4$
Clearly, $AB=4/3$ $⟹$ $PA+PB=AB$
$⟹P$ is any point on the line segment AB.
Ans: 1