Q47 / 79 · NEST (NISER) 2026 · Mathematics
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
The locus of $z$ satisfying $|z + \hat{z}| = |z - \hat{z}|$ in the complex plane is
- Correct answerAunion of the real axis and the imaginary axis
- Bthe real axis
- Cthe straight line $y = x$
- Da circle
Solution
Given $z = x + iy$ and $\hat{z} = y + ix$.
$z + \hat{z} = (x + y) + i(x + y)$
$|z + \hat{z}| = \sqrt{(x + y)^2 + (x + y)^2} = \sqrt{2}|x + y|$
$z - \hat{z} = (x - y) + i(y - x)$
$|z - \hat{z}| = \sqrt{(x - y)^2 + (y - x)^2} = \sqrt{2}|x - y|$
The given condition is $|z + \hat{z}| = |z - \hat{z}|$.
Substituting the magnitudes:
$\sqrt{2}|x + y| = \sqrt{2}|x - y|$
Squaring both sides:
$(x + y)^2 = (x - y)^2$
$x^2 + y^2 + 2xy = x^2 + y^2 - 2xy$
$4xy = 0 \Rightarrow xy = 0$
This implies $x = 0$ or $y = 0$.
$x = 0$ represents the imaginary axis and $y = 0$ represents the real axis.
Thus, the locus of $z$ is the union of the real axis and the imaginary axis.
$z + \hat{z} = (x + y) + i(x + y)$
$|z + \hat{z}| = \sqrt{(x + y)^2 + (x + y)^2} = \sqrt{2}|x + y|$
$z - \hat{z} = (x - y) + i(y - x)$
$|z - \hat{z}| = \sqrt{(x - y)^2 + (y - x)^2} = \sqrt{2}|x - y|$
The given condition is $|z + \hat{z}| = |z - \hat{z}|$.
Substituting the magnitudes:
$\sqrt{2}|x + y| = \sqrt{2}|x - y|$
Squaring both sides:
$(x + y)^2 = (x - y)^2$
$x^2 + y^2 + 2xy = x^2 + y^2 - 2xy$
$4xy = 0 \Rightarrow xy = 0$
This implies $x = 0$ or $y = 0$.
$x = 0$ represents the imaginary axis and $y = 0$ represents the real axis.
Thus, the locus of $z$ is the union of the real axis and the imaginary axis.
Hence, the correct answer is option: 'A'.