Question

How do you write the equation of a circle given two points $?$

Answer 1

Writing equation of the circle:

Suppose $A(x_1,y_1)$ and $B(x_2,y_2)$ be the two end points of diameter, then the equation of a circle with two points is written as:

$\mathbf{(}\mathit{x}\mathbf{}\mathbf{–}\mathbf{}{\mathit{x}}_{\mathbf{1}}\mathbf{)}\mathbf{(}\mathit{x}\mathbf{}\mathbf{–}\mathbf{}{\mathit{x}}_{\mathbf{2}}\mathbf{)}\mathbf{}\mathbf{+}\mathbf{}\mathbf{(}\mathit{y}\mathbf{}\mathbf{–}\mathbf{}{\mathit{y}}_{\mathbf{1}}\mathbf{)}\mathbf{(}\mathit{y}\mathbf{}\mathbf{–}\mathbf{}{\mathit{y}}_{\mathbf{2}}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$

Whose centre is $\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}$

Hence, by expanding these terms we can get the exact equation of a circle passing through the given two points.