Question

Area of the triangle formed by the complex numbers \(z\), \(iz\) and \(z+iz\) in the Argand diagram as vertices is:

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

Hint:

For area of triangle in Argand diagram, use \(\text{Area} = \frac{1}{2}|\text{Im}\{(z_2-z_1)\overline{(z_3-z_1)}\}|\).

Answer 1

The Correct Option is A

Step 1: Represent vertices.
The triangle vertices are:
\[ A = z, \quad B = iz, \quad C = z+iz \]

Step 2: Use area formula for complex numbers.
Area of triangle with vertices \(z_1, z_2, z_3\) is: \[ \text{Area} = \frac{1}{2} \left| \text{Im} \{ (z_2 - z_1) \overline{(z_3 - z_1)} \} \right| \]

Step 3: Compute differences.
\[ z_2 - z_1 = iz - z = (i-1)z, \quad z_3 - z_1 = (z+iz) - z = iz \]

Step 4: Compute product.
\[ (z_2 - z_1) \overline{(z_3 - z_1)} = (i-1)z \cdot \overline{iz} = (i-1)z \cdot (-i \overline{z}) = (i-1)(-i)|z|^2 \]

Step 5: Take imaginary part and magnitude.
\[ (i-1)(-i) = 1/2 \]
\(\text{Area} = \frac{1}{2}|z|^2\)

Step 6: Final conclusion.
\[ \boxed{\frac{1}{2}|z|^2} \]