Question

The area of the triangle in the complex plane formed by \( z, iz \) and \( z + iz \) is

• A. \( |z| \)
• B. \( |z|^2 \)
• C. \( \frac{1}{2}|z|^2 \)
• D. \( \frac{1}{2}|z + iz|^2 \)
• E. \( |z + iz| \)

Hint:

Area in complex plane problems often reduces to modulus squared when vectors involve rotations like \( iz \).

Answer 1

The Correct Option is C

Concept: Area of triangle formed by complex numbers \( z_1, z_2, z_3 \): \[ \text{Area} = \frac{1}{2} | \operatorname{Im}[(z_2 - z_1)\overline{(z_3 - z_1)}] | \]

Step 1: Take \( z \) as reference point.
\[ z_1 = z,\quad z_2 = iz,\quad z_3 = z + iz \] \[ z_2 - z_1 = iz - z = z(i - 1) \] \[ z_3 - z_1 = iz \]

Step 2: Apply formula.
\[ \text{Area} = \frac{1}{2} \left| \operatorname{Im}[(z(i-1)) \cdot \overline{iz}] \right| \]

Step 3: Compute conjugate.
\[ \overline{iz} = -i\bar{z} \] \[ (z(i-1))(-i\bar{z}) = |z|^2 (i-1)(-i) \]

Step 4: Simplify.
\[ (i-1)(-i) = 1 + i \] \[ \Rightarrow = |z|^2 (1 + i) \] Imaginary part: \[ = |z|^2 \]

Step 5: Final area.
\[ \text{Area} = \frac{1}{2}|z|^2 \]