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For all complex numbers z1 and z2 satisfying |z1| = 12 and |z2 - 3 - 4i| = 5, what is the minimum value of |z1 - z2|?

1. 0
2. 2
3. 7
4. 17

Correct answer: 2

Solution

The circle for z2 has centre at distance 5 from the origin, equal to its own radius, so it passes through the origin and lies entirely inside the circle for z1 (radius 12). The farthest point of z2's circle from the origin is 5+5=10, still inside radius 12. Minimum |z1-z2| = 12 - 10 = 2.

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