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
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
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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.
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.
Updated On: Jun 16, 2026
- union of the real axis and the imaginary axis
- the real axis
- a circle
- the straight line $y = x$
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
The question asks for the locus of a complex number $z$ satisfying a given modular relation.
We are given the definition of $\hat{z}$, which swaps the real and imaginary parts of $z$.
By substituting $z = x+iy$ and $\hat{z} = y+ix$, we can convert the equation into Cartesian coordinates.
Step 2: Key Formula or Approach:
The modulus of a complex number $w = a + ib$ (where $a, b \in \mathbb{R}$) is given by:
\[ |w| = \sqrt{a^2 + b^2} \]
Step 3: Detailed Explanation:
• Let us write down the expressions for $z + \hat{z}$ and $z - \hat{z}$:
\[ z + \hat{z} = (x + iy) + (y + ix) = (x + y) + i(y + x) = (x + y)(1 + i) \]
\[ z - \hat{z} = (x + iy) - (y + ix) = (x - y) + i(y - x) = (x - y)(1 - i) \]
• Now, we find the modulus of both terms:
\[ |z + \hat{z}| = |(x + y)(1 + i)| = |x + y| \cdot |1 + i| = |x + y| \sqrt{1^2 + 1^2} = |x + y|\sqrt{2} \]
\[ |z - \hat{z}| = |(x - y)(1 - i)| = |x - y| \cdot |1 - i| = |x - y| \sqrt{1^2 + (-1)^2} = |x - y|\sqrt{2} \]
• The given equation is:
\[ |z + \hat{z}| = |z - \hat{z}| \]
Substituting our simplified modulus expressions:
\[ |x + y|\sqrt{2} = |x - y|\sqrt{2} \implies |x + y| = |x - y| \]
• To solve this, we square both sides:
\[ (x + y)^2 = (x - y)^2 \]
\[ x^2 + 2xy + y^2 = x^2 - 2xy + y^2 \]
• Canceling out $x^2$ and $y^2$ from both sides:
\[ 4xy = 0 \implies xy = 0 \]
• The equation $xy = 0$ means:
\[ x = 0 \quad \text{or} \quad y = 0 \]
- $x = 0$ represents the imaginary axis.
- $y = 0$ represents the real axis.
Step 4: Final Answer:
The locus is the union of the real axis and the imaginary axis.
The question asks for the locus of a complex number $z$ satisfying a given modular relation.
We are given the definition of $\hat{z}$, which swaps the real and imaginary parts of $z$.
By substituting $z = x+iy$ and $\hat{z} = y+ix$, we can convert the equation into Cartesian coordinates.
Step 2: Key Formula or Approach:
The modulus of a complex number $w = a + ib$ (where $a, b \in \mathbb{R}$) is given by:
\[ |w| = \sqrt{a^2 + b^2} \]
Step 3: Detailed Explanation:
• Let us write down the expressions for $z + \hat{z}$ and $z - \hat{z}$:
\[ z + \hat{z} = (x + iy) + (y + ix) = (x + y) + i(y + x) = (x + y)(1 + i) \]
\[ z - \hat{z} = (x + iy) - (y + ix) = (x - y) + i(y - x) = (x - y)(1 - i) \]
• Now, we find the modulus of both terms:
\[ |z + \hat{z}| = |(x + y)(1 + i)| = |x + y| \cdot |1 + i| = |x + y| \sqrt{1^2 + 1^2} = |x + y|\sqrt{2} \]
\[ |z - \hat{z}| = |(x - y)(1 - i)| = |x - y| \cdot |1 - i| = |x - y| \sqrt{1^2 + (-1)^2} = |x - y|\sqrt{2} \]
• The given equation is:
\[ |z + \hat{z}| = |z - \hat{z}| \]
Substituting our simplified modulus expressions:
\[ |x + y|\sqrt{2} = |x - y|\sqrt{2} \implies |x + y| = |x - y| \]
• To solve this, we square both sides:
\[ (x + y)^2 = (x - y)^2 \]
\[ x^2 + 2xy + y^2 = x^2 - 2xy + y^2 \]
• Canceling out $x^2$ and $y^2$ from both sides:
\[ 4xy = 0 \implies xy = 0 \]
• The equation $xy = 0$ means:
\[ x = 0 \quad \text{or} \quad y = 0 \]
- $x = 0$ represents the imaginary axis.
- $y = 0$ represents the real axis.
Step 4: Final Answer:
The locus is the union of the real axis and the imaginary axis.
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