Question

Let a complex number be w = 1 - √3 i. Let another complex number z be such that |z w| = 1 and arg(z) - arg(w) = π/2. Then the area of the triangle with vertices origin, z and w is equal to :

• A. 1/2
• B. 2
• C. 1/4
• D. 4

Hint:

The area of a triangle with vertices $0, z_1, z_2$ is $\frac{1}{2} |z_1| |z_2| \sin(\theta_2 - \theta_1)$.

Answer 1

The Correct Option is A

Step 1: $|w| = \sqrt{1^2 + (-\sqrt{3})^2} = 2$. Given $|z w| = 1 \implies |z| \cdot |w| = 1 \implies |z| = 1/2$.
Step 2: Given $\arg(z) - \arg(w) = \pi/2$. This means the vectors $z$ and $w$ from the origin are perpendicular.
Step 3: The triangle is a right-angled triangle with origin as the vertex at the right angle. Area $= \frac{1}{2} \cdot |z| \cdot |w| = \frac{1}{2} \cdot \frac{1}{2} \cdot 2 = 1/2$.