Complex Numbers – Equation of a Circle

Complex Numbers – Equation of a Circle

Equation of a Circle: Consider a fixed complex number zₒ and let z be any complex number which moves in such a way that its distance from zₒ is always to r. this implies z would lie on a circle whose center is zₒ and radius is r. and its equation would be

|z – zₒ| = r

Squaring on both sides

|z – zₒ|² = r²



Let -a = zₒ and \({{z}_{0}}\overline{{{z}_{0}}}-{{r}^{2}}=b\). Then


It represents the general equation of a circle in the complex plane.

Now, let us consider a circle described on a lime segment AB \((A({{z}_{1}}),B({{z}_{2}}))\) as its diameter. Let P(z) be any point on the circle. As the angle in the semicircle is \(\frac{\pi }{2}\), So

\(\angle APB=\frac{\pi }{2}\)
Equation of a Circle

 \(\arg \left( \frac{{{z}_{1}}-z}{{{z}_{2}}-z} \right)=\pm \frac{\pi }{2}\),

\(\frac{z-{{z}_{1}}}{z-{{z}_{2}}}\) is purely imaginary