Question

Area of the triangle in the Argand diagram formed by the complex numbers $z$, $iz$, $z + iz$ where $z = x + iy$ is

• A. $|z|$
• B. $|z|^2$
• C. $2|z|^2$
• D. $\dfrac{1}{2}|z|^2$

Hint:

Multiplication by $i$ rotates a complex number by $90^\circ$ without changing its magnitude. Use this to identify perpendicular vectors in Argand diagrams.

Answer 1

The Correct Option is D

Step 1: Conceptual Understanding:
Multiplying a complex number by $i$ rotates it by $90^\circ$, so $z$ and $iz$ are perpendicular and have the same magnitude.
Step 2: Explanation in Detail:
$|z| = |iz|$ and they are perpendicular. The triangle with vertices at $0$, $z$, $iz$ is a right-angled isosceles triangle.
Area $= \dfrac{1}{2} \times |z| \times |iz| = \dfrac{1}{2}|z|^2$.
Step 3: Therefore, Stating the Final Answer
Area $= \dfrac{1}{2}|z|^2$.