The formula of a circle is as follows:
[tex] (x-h)^{2} + (y-k)^{2} = r^{2} [/tex]
[tex]h[/tex] and [tex]k[/tex] stand for the set of coordinates [tex](h, k)[/tex] which is the center of the circle, and [tex]r[/tex] is the size of the radius.
With the diameter given as [tex]A=(-2, 11), B=(6, 23)[/tex] We can find the middle by finding the middle of the x's and the y's. So [tex]h=\frac{-2+6}{2}, k=\frac{11+23}{2}[/tex]. This simplifies to [tex](h, k)=(2, 17)[/tex] so our formula currently looks as such:
[tex](x-2)^{2}+(y-17)^{2}=r^{2}[/tex]
Now finding r. Finding the length of a line from point A to point B is [tex] length=\sqrt{(A_{x}-B_{x})^{2}+(A_{y}-B_{y})^{2}} [/tex]
We can plug in our values for A and B as such...
[tex] \sqrt{(-2-6)^{2}+(11-23)^{2}} [/tex]
And this simplifies to the length of the diameter, 14.422. To get the radius, simply divide by 2: 7.211, and then square that to find what goes at the very end of the circle formula: 52.
This gives us a final formula of [tex](x-2)^{2}+(y-17)^{2}=52[/tex]
