To find the equation of the circle given the endpoints of its diameter, follow these steps:
1. Determine the center of the circle:
The center of the circle is the midpoint of the diameter. The midpoint of two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
For points [tex]\(P = (-2, -1)\)[/tex] and [tex]\(Q = (2, 1)\)[/tex], the coordinates of the midpoint (center) are:
[tex]\[
\left( \frac{-2 + 2}{2}, \frac{-1 + 1}{2} \right) = \left( \frac{0}{2}, \frac{0}{2} \right) = (0.0, 0.0)
\][/tex]
Thus, the center of the circle is [tex]\((0.0, 0.0)\)[/tex].
2. Calculate the radius of the circle:
The radius of the circle is half the length of the diameter. The length of the diameter can be found using the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Using points [tex]\(P = (-2, -1)\)[/tex] and [tex]\(Q = (2, 1)\)[/tex], the distance (diameter) is:
[tex]\[
\sqrt{(2 - (-2))^2 + (1 - (-1))^2} = \sqrt{(2 + 2)^2 + (1 + 1)^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20}
\][/tex]
The radius [tex]\(r\)[/tex] is half of this distance:
[tex]\[
r = \frac{\sqrt{20}}{2} \approx \sqrt{5.000000000000001}
\][/tex]
3. Formulate the equation of the circle:
The general 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]
For our circle, the center [tex]\((h, k)\)[/tex] is [tex]\((0.0, 0.0)\)[/tex] and the radius [tex]\(r^2\)[/tex] is approximately [tex]\(5.000000000000001\)[/tex]. Therefore, the equation of the circle is:
[tex]\[
(x - 0.0)^2 + (y - 0.0)^2 = 5.000000000000001
\][/tex]
Simplifying, we get:
[tex]\[
x^2 + y^2 = 5.000000000000001
\][/tex]
Final answer in the form of [tex]\((x-[?])^2 +(y-[])^2=[]\)[/tex]:
[tex]\[
\boxed{(0.0), (0.0), 5.000000000000001}
\][/tex]
