Question

Find the equation of a circle with centers (1, 1) and contains points (2, 3)

Answer 1

Equation of a circle centered at $(1,1)$ with radius $R$ is $(x-1)^2+(y-1)^2=R^2$. Now to find the radius, we can just use the fact that $(2,3)$ lies on it :

$(2-1)^2 + (3-1)^2 = 5=R^2$

Therefore the equation is $(x-1)^2+(y-1)^2=5$ (or in complex number notation $|z-1-i|=\sqrt{5}$).