Question

What is the area of a circle passing through $z_{1} = \frac{5}{2} + \frac{\sqrt{27i}}{2}$ and having center $z_{2} = e^{\frac{\pi }{3}i}$

Answer 1

The distance between z₁ and z₂ corresponds to the radius of the circle.

|z₁ - z₂| = |(5/2 + i √27/2) - exp(iπ/3)|

… = |(5/2 + i √27/2) - (cos(π/3) + i sin(π/3))|

… = |(5/2 + i √27/2) - (1/2 + i √3/2)|

… = |2 + i √3|

… = √(2² + (√3)²)

… = √(4 + 3) = √7

Then the area of the circle is π (√7)² = 7π.