To determine the equation of circle [tex]\( M \)[/tex] with endpoints [tex]\( F(2, 6) \)[/tex] and [tex]\( G(14, 22) \)[/tex] forming the diameter, follow these steps:
1. Find the center of the circle:
The center of the circle is the midpoint of the line segment joining the points [tex]\( F \)[/tex] and [tex]\( G \)[/tex]. The formula for the midpoint [tex]\((x_m, y_m)\)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}
\][/tex]
Plugging in the coordinates of points [tex]\( F \)[/tex] and [tex]\( G \)[/tex]:
[tex]\[
x_m = \frac{2 + 14}{2} = 8.0
\][/tex]
[tex]\[
y_m = \frac{6 + 22}{2} = 14.0
\][/tex]
So, the center of the circle is [tex]\((8, 14)\)[/tex].
2. Find the radius of the circle:
The radius is half the length of the diameter. First, find the length of the diameter using the distance formula:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting the coordinates of points [tex]\( F \)[/tex] and [tex]\( G \)[/tex]:
[tex]\[
d = \sqrt{(14 - 2)^2 + (22 - 6)^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20.0
\][/tex]
Since the radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
r = \frac{20.0}{2} = 10.0
\][/tex]
3. Write the equation of the circle:
The standard form of the 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 the center [tex]\((8, 14)\)[/tex] and radius [tex]\(10.0\)[/tex]:
[tex]\[
(x - 8)^2 + (y - 14)^2 = 10^2
\][/tex]
[tex]\[
(x - 8)^2 + (y - 14)^2 = 100
\][/tex]
So, the equation of circle [tex]\( M \)[/tex] is:
[tex]\[
(x-8)^2+(y-14)^2=100
\][/tex]
