If z is a complex number satisfying |z-3+4i|=2, then the difference between the greatest and the least values of |z|^2 is

The condition $|z-(3-4i)|=2$ describes a circle with centre $C=(3,-4)$ and radius $2$. The distance of the centre from the origin is $|OC|=\sqrt{9+16}=5$. For points on this circle, the greatest value of $|z|$ is $|OC|+2=7$ and the least value is $|OC|-2=3$. Therefore the greatest value of $|z|^2$ is $49$ and the least is $9$, so the required difference is $49-9=40$. A common slip is to subtract the radii-based quantities directly (giving $4$ or $20$) instead of squaring the extreme distances first.