Complex Numbers
Modulus Identity on Unit Circle
Complex Numbers_PYQ
Grade 11
Question:
If $z_1,z_2$ and $z_3$ are complex numbers such that $|z_1|=|z_2|=|z_3|=\left|\dfrac{1}{z_1}+\dfrac{1}{z_2}+\dfrac{1}{z_3}\right|=1$, then $|z_1+z_2+z_3|$ is
equal to $1$
less than $1$
greater than $3$
equal to $3$
Step-by-Step Solution
Key Concept: On the unit circle $1/z_k=\bar{z}_k$, so the sum of reciprocals is the conjugate of the sum. Since $|\bar{w}|=|w|$, both sums have the same modulus.
**Step 1: Convert reciprocals using |z|=1**
Since $|z_k|=1$, $\bar{z}_k=\dfrac{1}{z_k}$. Therefore $\dfrac{1}{z_1}+\dfrac{1}{z_2}+\dfrac{1}{z_3}=\bar{z}_1+\bar{z}_2+\bar{z}_3=\overline{z_1+z_2+z_3}$.
**Step 2: Conclude**
$\left|\dfrac{1}{z_1}+\dfrac{1}{z_2}+\dfrac{1}{z_3}\right|=|\overline{z_1+z_2+z_3}|=|z_1+z_2+z_3|=1$.
Correct Answer: 1