The center of the circle |z - (a + ib)| = r has Cartesian coordinates (a, b). Which of the following is the center in the complex plane?

The center of the circle |z - (a + ib)| = r has Cartesian coordinates (a, b). Which of the following is the center in the complex plane?

• A. a + ib
• B. a - ib
• C. -a - ib
• D. b + ia

Answer: (A)

In the complex plane, a complex number z = x + iy corresponds to the point (x, y). The circle |z − z0| = r has center at z0. If the center is given to have Cartesian coordinates (a, b), that center is the complex number a + ib (real part a, imaginary part b). So the center in the complex plane is a + ib. The other expressions would place the center at points with different coordinates: a − ib is (a, −b), −a − ib is (−a, −b), and b + ia is (b, a). None of these match (a, b) in general, so they’re not the center.

In the complex plane, a complex number z = x + iy corresponds to the point (x, y). The circle |z − z0| = r has center at z0. If the center is given to have Cartesian coordinates (a, b), that center is the complex number a + ib (real part a, imaginary part b). So the center in the complex plane is a + ib.

The other expressions would place the center at points with different coordinates: a − ib is (a, −b), −a − ib is (−a, −b), and b + ia is (b, a). None of these match (a, b) in general, so they’re not the center.