For the circle |z - (a+ib)| = r, what is the radius?

For the circle |z - (a+ib)| = r, what is the radius?

• A. r
• B. a
• C. sqrt(a^2 + b^2)
• D. b

Answer: (A)

In the complex plane, a circle with center at c and radius R is described by the set of points z with |z − c| = R. Here the center is a + ib, so |z − (a + ib)| measures the distance from z to that center. Since all points on the circle are at this fixed distance R from the center, that fixed distance is the radius. Therefore, the radius is r. The values a and b only determine where the center sits; they don’t change the size of the circle. By contrast, sqrt(a^2 + b^2) would be the distance from the origin to the center, not the radius.

In the complex plane, a circle with center at c and radius R is described by the set of points z with |z − c| = R. Here the center is a + ib, so |z − (a + ib)| measures the distance from z to that center. Since all points on the circle are at this fixed distance R from the center, that fixed distance is the radius. Therefore, the radius is r. The values a and b only determine where the center sits; they don’t change the size of the circle. By contrast, sqrt(a^2 + b^2) would be the distance from the origin to the center, not the radius.