Question

1. Let \( z_{1}, z_{2} \in C \) such that \( \left|z_{1}+z_{2}\right|=\sqrt{3} \) and \( \left|z_{1}\right|=\left|z_{2}\right|=1 \) then the value of \( \left|z_{1}-z_{2}\right| \) is

Answer 1

We have |z₁ + z₂| = √3

and |z₁| = |z₂| = 1

∵ |z₁ + z₂|² + |z₁ - z₂|² = 2(|z₁|² + |z₂|²)

⇒ |z₁ - z₂|² = 2(|z₁|² + |z₂|²) - |z₁+z₂|²

= 2(1+1) - 3  (∵ |z₁| = |z₂| = 1 and |z₁ + z₂| = √3)

= 4 - 3 = 1