Answer:
[tex](x-3)^{2} +(y-2)^{2}=41[/tex]
Step-by-step explanation:
step 1
Find the diameter of the circle
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
we have
[tex]A(8,-2)\\E(-2,6)[/tex]
substitute the values
[tex]d=\sqrt{(6+2)^{2}+(-2-8)^{2}}[/tex]
[tex]d=\sqrt{(8)^{2}+(-10)^{2}}[/tex]
[tex]d=\sqrt{164}\ units[/tex]
[tex]d=2\sqrt{41}\ units[/tex]
step 2
Find the center of the circle
The center is the midpoint of the diameter
The center is equal to
[tex]C=(\frac{8-2}{2},\frac{-2+6}{2})[/tex]
[tex]C=(3,2)[/tex]
step 3
Find the equation of the circle
The equation of the circle in center radius form is equal to
[tex](x-h)^{2} +(y-k)^{2}=r^{2}[/tex]
we have
(h,k)=(3,2)
[tex]r=2\sqrt{41}/2=\sqrt{41}\ units[/tex] ---> the radius is half the diameter
substitute
[tex](x-3)^{2} +(y-2)^{2}=(\sqrt{41})^{2}[/tex]
[tex](x-3)^{2} +(y-2)^{2}=41[/tex]
