The circle equation |z - (a + ib)| = r has its centre at which point?

The circle equation |z - (a + ib)| = r has its centre at which point?

• A. a + ib
• B. a - ib
• C. 0
• D. 1

Answer: (A)

The distance interpretation is key here: in the complex plane, the modulus |z − z0| is the distance from z to the fixed point z0. So |z − (a + ib)| = r is the set of all points z whose distance to the fixed point a + ib is r. That means the circle is centered at a + ib with radius r. If you write z as x + iy, you get sqrt[(x − a)^2 + (y − b)^2] = r, which shows the center is at the point (a, b) in the plane, i.e., the complex number a + ib.

The distance interpretation is key here: in the complex plane, the modulus |z − z0| is the distance from z to the fixed point z0. So |z − (a + ib)| = r is the set of all points z whose distance to the fixed point a + ib is r. That means the circle is centered at a + ib with radius r. If you write z as x + iy, you get sqrt[(x − a)^2 + (y − b)^2] = r, which shows the center is at the point (a, b) in the plane, i.e., the complex number a + ib.