JEE Main 2025 — Complex Number Question with Solution

Let $z=x+iy\Rightarrow \bar{z}=x-iy$ $\begin{aligned} & 3|\bar{z}-i|=1|2 \bar{z}+i| \\ & =3 \mid(x-(y+1) i|=|2 x+i(1-2 y)| \\ & =3 \sqrt{x^2+(y+1)^2}=\sqrt{(2 x)^2+(1-2 y)^2} \\ & =9\left(x^2+y^2+2 y+1\right)=4 x^2+4 y^2-4 y+1 \\ & \Rightarrow 5 x^2+5 y^2+22 y+8=0 \\ & \Rightarrow \text { Centre } \equiv\left(0,-\frac{11}{5}\right) \end{aligned}$
Area of $\Delta$ $=\frac{1}{2}|\alpha |\left|\frac{-11}{5}\right|=11$ $\begin{array}{rl}& \Rightarrow |\alpha |=10\\ & \Rightarrow {\alpha }^2=100\end{array}$