Question

If $|z+\bar{z}|=|z-\bar{z}|$, then the locus of $z$ is

• A. a pair of straight lines
• B. a rectangular hyperbola
• C. a line
• D. a set of four lines

Answer 1

Answer: (A)

Let $z=x+iy$. Then,
$|z+\bar{z}|=|z-\bar{z}|$
$\iff |2x|=|2iy|$
$\Rightarrow |x|=|y|\iff x=\pm y$
Hence, the locus of $z$ is a pair of straight lines.