Let z be a complex number such that |z|=1. If (2+ k^2 z)/(k +z̄)= k z, k ∈ R, then the maximum distance of k +i k^2 from the circle |z-(1+2 i)|=1 is:

Let $\displaystyle z$ be a complex number such that $\displaystyle |z|=1$. If $\displaystyle \frac{2+\mathrm{k}^2 z}{\mathrm{k}+\bar{z}}=\mathrm{k} z, \mathrm{k} \in \mathbf{R}$, then the maximum distance of $\displaystyle \mathrm{k}+i \mathrm{k}^2$ from the circle $\displaystyle |z-(1+2 i)|=1$ is :