Cue Column:

• What are the two forms?
• How to find max/min of |z|?
• How to find max/min of |z - z0|?

Note Column: Form 1: |z1 + z2| <= |z1| + |z2| (upper bound) Form 2: |z1 + z2| >= ||z1| - |z2|| (lower bound)

Max/Min of |z| given |z - z0| = r: z lies on a circle centered at z0 with radius r.

• max|z| = |z0| + r (point on circle farthest from origin)
• min|z| = |z0| - r if |z0| > r, else 0 (if origin is inside the circle)

Max/Min of |z - a| given |z - b| = r: This is the distance from a to a point on the circle centered at b.

• max|z - a| = |a - b| + r
• min|z - a| = ||a - b| - r|

Important: For "find max of |z + 3 - 4i| given |z - 1 + 2i| <= 3", the constraint is a disk. Max occurs on the boundary circle, at the point farthest from -3+4i.

Summary: Triangle inequality gives bounds. Geometric interpretation: distances to points from circles.