Question

Let $z_{1}$ and $z_{2}$ be two non real complex cube roots of unity and $\left|z-z_{1}\right|^{2}+\left|z-z^{2}\right|^{2}=\lambda$ be the equation of a circle with $z_{1}, z_{2}$ as ends of a diameter, then the value of $\lambda$ is

• A. 4
• B. 3
• C. 2
• D. $\sqrt{2}$

Answer 1

Answer: (B)

We have,
$|z-\omega|^{2}+\left|z-\omega^{2}\right|^{2}=\lambda$
$\Rightarrow \lambda=\left|\omega-\omega^{2}\right|^{2}=\left|\omega^{2}+\omega^{4}-2 \omega^{3}\right|$
$=\left|\omega^{2}+\omega-2\right|=|-1-2|=3$