Question

Find the equation of a circle when the 2 end points of a diameter is given by A(-2, 1) , B(-1, 2)

Answer 1

The center of the circle has the coordinates: $O(-\frac32,\frac32)$. It is the midpoint of the segment A(-2,1), B(-1,2). The radius of the circle is: $r=\sqrt{(-\frac32+2)^2+(\frac32-1)^2}=\sqrt{\frac12}=\frac{\sqrt{2}}{2}$ . The equation is $(x+\frac32)^2+(y-\frac32)^2=\frac12$ .

We can also consider the problem with complex numbers. I.e., we may present points in the following form : A:-2+i; B:-1+2i and O: $-\frac32+\frac32i$. The radius is $r=\frac{\sqrt{2}}{2}$ (the distance between O and A). The equation of the circle is: $|z-z_0|^2=\frac12$ , where $z$ presents a complex variable. I.e., $z=x+iy$ and $z_0=-\frac32+\frac32i$