Question

If $z$ is a complex number such that $|3z-2|+|3z+2|=4$, then the locus of $z$ is

• A. a line
• B. a circle
• C. a point
• D. a line segment

Answer 1

The correct option is D a line segment

$|3z-2|+|3z+2|=4\Rightarrow |z-\frac{2}{3}|+|z+\frac{2}{3}|=\frac{4}{3}$
Let $A=(\frac{2}{3},0),B=(-\frac{2}{3},0)$
Then
$|PA|+|PB|=|AB|$
Hence, the locus of $P$ is the line segment $AB.$