JEE Main · 2021 · 31 Aug 2021 · Shift 2 · Complex Numbers

The ratio $\frac{z-i}{z-1}$ is purely imaginary when $z$ subtends a right angle on the segment from $1$ to $i$, so $z$ lies on the circle having that segment as diameter: centre $\left(\frac{1}{2},\frac{1}{2}\right)$ and radius $\frac{1}{\sqrt{2}}$. The distance from $(3,3)$ to the centre is $\sqrt{{\left(\frac{5}{2}\right)}^2+{\left(\frac{5}{2}\right)}^2}=\frac{5\sqrt{2}}{2}$, so the minimum of $|z-(3+3i)|$ is $\frac{5\sqrt{2}}{2}-\frac{1}{\sqrt{2}}=2\sqrt{2}$.