Greatest value

Let complex number $z_1+z_2$ be represented by the line $OP$ , where $O$ is the origin. Then the locus of $z_1+z_2$ or $P$ is a circle with centre $C$ at $(24,7)$ , the end of $z_1$ . Then $|z_1+z_2|$ is equal to the length of $OP$ . The longest $OP$ is when $P$ , $C$ and $O$ are colinear. Then the maximum $|z_1+z_2| = |z_1| + |z_2| = \sqrt{24^2+7^2} + 6 = 25+6 = \boxed{31}$ .