Question:
Area of the triangle in the Argand diagram formed by the complex numbers $z$, $iz$, $z + iz$ where $z = x + iy$ is
Area of the triangle in the Argand diagram formed by the complex numbers $z$, $iz$, $z + iz$ where $z = x + iy$ is
Show Hint
Multiplication by $i$ rotates a complex number by $90^\circ$ without changing its magnitude. Use this to identify perpendicular vectors in Argand diagrams.
Updated On: May 2, 2026
- $|z|$
- $|z|^2$
- $2|z|^2$
- $\dfrac{1}{2}|z|^2$
Show Solution
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept:
Multiplying a complex number by $i$ rotates it by $90^\circ$, so $z$ and $iz$ are perpendicular and have the same magnitude.
Step 2: Detailed Explanation:
$|z| = |iz|$ 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: Final Answer:
Area $= \dfrac{1}{2}|z|^2$.
Multiplying a complex number by $i$ rotates it by $90^\circ$, so $z$ and $iz$ are perpendicular and have the same magnitude.
Step 2: Detailed Explanation:
$|z| = |iz|$ 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: Final Answer:
Area $= \dfrac{1}{2}|z|^2$.
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