Question

What is the area of a circle passing through
`z_(1) = (5)/(2) + (√(27i))/(2)` and having center
`z_(2) = e^{(\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.