JEE 2025 — Mathematics PYQ

$\frac{2+k^{2}z}{k+z}=kz$

$2+k^{2}z=kz(k+z)$

$2+k^{2}z=k^{2}z+kz\bar{z}$

$|z{|}^2=z\bar{z}(\because |z|=1)$

$\Rightarrow 2=k\times z\bar{z}$

$\Rightarrow k=2$

$k+i{k}^2=2+4i$

$P(2,4)$

$|z-(1+2i)|=1$

$\text{Centre of circle }=(1,2)$

$\text{Radius }(r)=1$

$\text{Maximum distance of }P\text{ from the circle }=OP+r$

$OP+r=\sqrt{1^2+2^2}+1=\sqrt{5}+1$