JEE 2023 — Mathematics PYQ

\begin{aligned}
 & \left|z-z_{1}\right|^{2}+\left|z-z_{2}\right|^{2}=\left|z_{1}-z_{2}\right|^{2} \\
 & \mathrm{Let}z_{1}=\alpha,z_{2}=\beta \\
 & \Rightarrow|\alpha-\beta|^{2}=2\lambda \\
 & \Rightarrow\mid\alpha-\beta\mid=\sqrt{2\lambda} \\
 & \Rightarrow2r=\sqrt{2\lambda} & \because r={\sqrt{\lambda-1}} \\
 & \Rightarrow2\sqrt{\lambda-1}=\sqrt{2\lambda} \\
 & \lambda=2 \\
 & \Rightarrow|\alpha-\beta|=2
\end{aligned}