Question

For a complex number $z = x + iy$, where $x, y \in \mathbb{R}$, denote $\hat{z} = y + ix$. The locus of $z$ satisfying $|z + \hat{z}| = |z - \hat{z}|$ in the complex plane is

• A. union of the real axis and the imaginary axis
• B. the real axis
• C. a circle
• D. the straight line $y = x$

Hint:

Notice that $\hat{z}$ is related to the conjugate of $z$ by $\hat{z} = i\bar{z}$.
Substituting this gives $|z + i\bar{z}| = |z - i\bar{z}|$.
Squaring this and using properties of complex numbers leads to the same relation $xy = 0$ very quickly.

Answer 1

The Correct Option is A

Step 1: Write $z$ and $\hat{z}$ in components.
Take $z = x + iy$. By the given rule, $\hat{z} = y + ix$, which simply swaps the real and imaginary parts.

Step 2: Form the sum $z + \hat{z}$.
Adding, $z + \hat{z} = (x + y) + i(y + x) = (x + y) + i(x + y)$. Both parts equal $x + y$.

Step 3: Form the difference $z - \hat{z}$.
Subtracting, $z - \hat{z} = (x - y) + i(y - x) = (x - y) - i(x - y)$. Both parts have size $|x - y|$.

Step 4: Compute the two moduli.
$|z + \hat{z}| = \sqrt{(x + y)^2 + (x + y)^2} = |x + y|\sqrt{2}$. Similarly $|z - \hat{z}| = |x - y|\sqrt{2}$.

Step 5: Apply the given equality.
Setting them equal and dividing by $\sqrt{2}$: $|x + y| = |x - y|$. Squaring both sides gives $(x + y)^2 = (x - y)^2$.

Step 6: Simplify to find the locus.
Expanding, $x^2 + 2xy + y^2 = x^2 - 2xy + y^2$, so $4xy = 0$, meaning $xy = 0$. That holds when $x = 0$ or $y = 0$, which is exactly the imaginary axis together with the real axis.
\[ \boxed{\text{the union of the real axis and the imaginary axis}} \]