QJEEM24Mae145: JEE Main 2024 Mathematics - Class 11 JEEM24Mae145 Solution: Let the complex numbers and lie on the circles and…

$\begin{aligned}& \left|z-z_0\right|^2=4 \\& \Rightarrow\left(\alpha-z_0\right)\left(\bar{\alpha}-\bar{z}_0\right)=4 \\& \Rightarrow \alpha \bar{\alpha}-\alpha \bar{z}_0-z_0 \bar{\alpha}+\left|z_0\right|^2=4 \\& \Rightarrow|\alpha|^2-\alpha \bar{z}_0-z_0 \bar{\alpha}=2 \quad\text{......... (1)} \\& \left|z-z_0\right|^2=16\end{aligned}$

$\begin{aligned}& \Rightarrow\left(\frac{1}{\bar{\alpha}}-z_0\right)\left(\frac{1}{\alpha}-\bar{z}_0\right)=16 \\& \Rightarrow\left(1-\bar{\alpha} z_0\right)\left(1-\alpha \bar{z}_0\right)=16|\alpha|^2 \\& \Rightarrow 1-\bar{\alpha} z_0-\alpha \bar{z}_0+|\alpha|^2\left|z_0\right|^2=16|\alpha|^2 \\& \Rightarrow 1-\bar{\alpha} z_0-\alpha \bar{z}_0=14|\alpha|^2 \quad\text{....... (2)}\end{aligned}$

From (1) and (2)

$\begin{aligned}& \Rightarrow 5|\alpha|^2=1 \\& \Rightarrow 100|\alpha|^2=20\end{aligned}$