Complex Numbers
Modulus and Argument
Grade Class 11
Question:
<p>If \( |z_1| = |z_2| = |z_3| = \left|\dfrac{1}{z_1} + \dfrac{1}{z_2} + \dfrac{1}{z_3}\right| = 1 \), then \( |z_1 + z_2 + z_3| \) is:</p>
Equal to 1
Less than 1
Equal to 3
Greater than 1
Step-by-Step Solution
Key Concept: Since |zₖ|=1, 1/zₖ = z̄ₖ. So |z̄_1+z̄_2+z̄_3|=1 \Rightarrow |z_1+z_2+z_3|=1.
<p>$ |z_k|=1 \Rightarrow \dfrac{1}{z_k} = \bar{z}_k $. So $ \left|\dfrac{1}{z_1}+\dfrac{1}{z_2}+\dfrac{1}{z_3}\right| = |\bar{z}_1+\bar{z}_2+\bar{z}_3| = |\overline{z_1+z_2+z_3}| = |z_1+z_2+z_3| = 1 $.</p>
Correct Answer: C