To find the equation of the circle given the endpoints [tex]\( F(2,6) \)[/tex] and [tex]\( G(14,22) \)[/tex] of its diameter, we will follow these steps:
1. Find the center of the circle:
The center of the circle is the midpoint of the endpoints of the diameter. The formula for the midpoint [tex]\((x, y)\)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Plugging in the coordinates of [tex]\(F\)[/tex] and [tex]\(G\)[/tex]:
[tex]\[
\left( \frac{2 + 14}{2}, \frac{6 + 22}{2} \right) = (8.0, 14.0)
\][/tex]
Therefore, the center of the circle is [tex]\((8.0, 14.0)\)[/tex].
2. Calculate 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. For points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], the distance [tex]\( d \)[/tex] is:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Plugging in the coordinates of [tex]\(F\)[/tex] and [tex]\(G\)[/tex]:
[tex]\[
d = \sqrt{(14 - 2)^2 + (22 - 6)^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20
\][/tex]
The radius is half of the diameter:
[tex]\[
\text{Radius} = \frac{20}{2} = 10.0
\][/tex]
3. Formulate 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]
Here, the center is [tex]\((8.0, 14.0)\)[/tex] and the radius is [tex]\(10.0\)[/tex]. Therefore:
[tex]\[
(x - 8.0)^2 + (y - 14.0)^2 = 10.0^2
\][/tex]
Simplifying [tex]\( 10.0^2 \)[/tex]:
[tex]\[
(x - 8.0)^2 + (y - 14.0)^2 = 100.0
\][/tex]
Thus, the equation of circle [tex]\(M\)[/tex] is:
[tex]\[
(x - 8)^2 + (y - 14)^2 = 100
\][/tex]
