If $z$ is a complex number such that $|z-3|+|z+3|=8$, then the maximum value of $|z|$ is

Solution

1. The equation $|z-3|+|z+3|=8$ represents an ellipse in the complex plane with foci at $z_1=3$ and $z_2=-3$.
2. The sum of the distances from any point $z$ to the foci is constant and equal to the length of the major axis, $2a = 8$, which gives $a=4$.
3. The center of this ellipse is at the origin $(0,0)$, and the semi-major axis length $a=4$ represents the maximum distance from the center to any point on the ellipse.
4. Hence the answer is (B).