If z is a complex number with |z-3-4i|≤2, then the maximum value of |z|^2 is

The condition says $z$ lies in the closed disc with centre $C(3,4)$ and radius $2$. The distance of the centre from the origin is $|3+4i|=5$. The point of the disc farthest from the origin lies on the ray from $O$ through $C$, at distance $5+2=7$ from $O$ (the nearest would be $5-2=3$). Hence $\max|z|=7$ and $\max|z|^2=49$. By the triangle inequality, $|z|\le|z-3-4i|+|3+4i|\le2+5=7$, with equality attainable at $z=3+4i+\tfrac25(3+4i)$.