To write the equation of the circle given that points [tex]\( F(2,6) \)[/tex] and [tex]\( G(14,22) \)[/tex] are the endpoints of a diameter, we proceed with the following steps:
1. Determine the center of the circle:
The center of the circle, denoted by [tex]\((h, k)\)[/tex], is the midpoint of the diameter formed by points [tex]\( F \)[/tex] and [tex]\( G \)[/tex]. To find the midpoint, we use the midpoint formula:
[tex]\[
h = \frac{x_1 + x_2}{2}, \quad k = \frac{y_1 + y_2}{2}
\][/tex]
Substituting the coordinates:
[tex]\[
h = \frac{2 + 14}{2} = 8, \quad k = \frac{6 + 22}{2} = 14
\][/tex]
So, the center of the circle is [tex]\((8, 14)\)[/tex].
2. Determine the radius of the circle:
The radius [tex]\( r \)[/tex] of the circle is half the length of the diameter. To find the length of the diameter, we use the distance formula:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting the coordinates:
[tex]\[
d = \sqrt{(14 - 2)^2 + (22 - 6)^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20
\][/tex]
Hence, the radius is:
[tex]\[
r = \frac{d}{2} = \frac{20}{2} = 10
\][/tex]
3. Form the equation of the circle:
The standard equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\( r \)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Substituting the values we found for [tex]\( h \)[/tex], [tex]\( k \)[/tex], and [tex]\( r \)[/tex]:
[tex]\[
(x - 8)^2 + (y - 14)^2 = 10^2
\][/tex]
Simplifying the radius squared:
[tex]\[
(x - 8)^2 + (y - 14)^2 = 100
\][/tex]
Therefore, the equation of the circle is:
[tex]\[
(x - 8)^2 + (y - 14)^2 = 100
\][/tex]
