Question

If \( z = x + iy \), \( xy \neq 0 \), satisfies the equation \( z^2 + i\overline{z} = 0 \), then \( |z|^2 \) is equal to:

• A. 9
• B. 1
• C. 4
• D. \( \frac{1}{4} \)

Answer 1

The Correct Option is B

Given: \[ z^2 + i\overline{z} = 0 \] where \( z = x + iy \) and \( \overline{z} = x - iy \). 
Substitute \( z = x + iy \): \[ z^2 = (x + iy)^2 = x^2 - y^2 + 2ixy \] and \[ i\overline{z} = i(x - iy) = ix + y \]  
Substitute into the equation: \[ (x^2 - y^2 + 2ixy) + (ix + y) = 0 \] 
Separate the real and imaginary parts: 
From the imaginary part: \[ x(2y + 1) = 0 \] 
Since \( x \neq 0 \), we have \( 2y + 1 = 0 \Rightarrow y = -\frac{1}{2} \).  
Substitute \( y = -\frac{1}{2} \) into the real part: \[ x^2 - \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) = 0 \] \[ x^2 - \frac{1}{4} - \frac{1}{2} = 0 \] \[ x^2 = \frac{3}{4} \Rightarrow x = \pm \frac{\sqrt{3}}{2} \]  
Calculate \( |z|^2 \): \[ |z|^2 = x^2 + y^2 = \left(\frac{\sqrt{3}}{2}\right)^2 + \left(-\frac{1}{2}\right)^2 = \frac{3}{4} + \frac{1}{4} = 1 \] \[ |z|^2 = 1 \]

Answer 2

To determine the value of \( |z|^2 \) where \( z = x + iy \) and it satisfies the equation \( z^2 + i\overline{z} = 0 \), let's proceed with a step-by-step solution.

Step 1: Express the Given Equation

The complex number \( z \) is represented as \( z = x + iy \), where \( x \) and \( y \) are real numbers. The conjugate of \( z \) is given by \( \overline{z} = x - iy \).

Substitute these into the given equation:

\(z^2 + i\overline{z} = (x + iy)^2 + i(x - iy) = 0\)

Now, compute the square of \( z \):

\((x + iy)^2 = x^2 - y^2 + 2ixy\)

Substitute back into the equation:

\(x^2 - y^2 + 2ixy + ix - y = 0\)

Step 2: Separate Real and Imaginary Parts

From the expression above, separate the real and imaginary parts:

• Real Part: \(x^2 - y^2 - y = 0\)
• Imaginary Part: \(2xy + x = 0\)

Step 3: Solve the System of Equations

From the imaginary part, solve for \( x \):

\(x(2y + 1) = 0\)

This yields \( x = 0 \) or \( 2y + 1 = 0 \). However, if \( x = 0 \), then \( z = iy \), which cannot satisfy \( xy \neq 0 \). Hence, solve for \( y \):

\(2y + 1 = 0 \implies y = -\frac{1}{2}\)

Substitute \( y = -\frac{1}{2} \) into the real part equation:

\(x^2 - \left(-\frac{1}{2}\right)^2 - \left(-\frac{1}{2}\right) = 0\)

This simplifies to:

\(x^2 - \frac{1}{4} + \frac{1}{2} = 0 \implies x^2 = \frac{1}{4}\)

Thus, \( x = \pm \frac{1}{2} \).

Step 4: Calculate \( |z|^2 \)

Now, compute the magnitude squared, \( |z|^2 = x^2 + y^2 \):

\(|z|^2 = \left(\pm \frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right)^2 = \frac{1}{4} + \frac{1}{4} = \frac{1}{2} + \frac{1}{2} = 1\)

Thus, the value of \( |z|^2 \) is \( 1 \).

Conclusion: The value of \( |z|^2 \), given that \( z = x + iy \) satisfies the equation \( z^2 + i\overline{z} = 0 \), is: 1.