Let w=z z̄+k_1 z+k_2 i z+λ(1+i), k_1, k_2 ∈ R. Let Re (w)=0 be the circle C of radius 1 in the first quadrant touching the line y=1 and the y -axis.…

Let $\displaystyle w=z \bar{z}+k_1 z+k_2 i z+\lambda(1+i), k_1, k_2 \in \mathbb{R}$. Let $\displaystyle \operatorname{Re}(w)=0$ be the circle C of radius $\displaystyle 1$ in the first quadrant touching the line $\displaystyle y=1$ and the $\displaystyle y$-axis. If the curve $\displaystyle \operatorname{Im}(w)=0$ intersects C at A and B , then $\displaystyle 30(A B)^2$ is equal to $\displaystyle \_\_\_\_$.