To find the equation of the circle given that [tex]\(P = (-2, -1)\)[/tex] and [tex]\(Q = (2, 1)\)[/tex] are the endpoints of its diameter, follow these detailed steps:
1. Find the center of the circle: The center of the circle is the midpoint of the diameter.
Midpoint formula is:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Plugging in the coordinates of [tex]\(P\)[/tex] and [tex]\(Q\)[/tex]:
[tex]\[
\text{Center} = \left( \frac{-2 + 2}{2}, \frac{-1 + 1}{2} \right) = (0, 0)
\][/tex]
2. Calculate the radius of the circle: The radius is half the length of the diameter. First, find the distance between [tex]\(P\)[/tex] and [tex]\(Q\)[/tex], which represents the length of the diameter.
Use the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting the given coordinates:
[tex]\[
\text{Distance} = \sqrt{(2 - (-2))^2 + (1 - (-1))^2} = \sqrt{(2 + 2)^2 + (1 + 1)^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}
\][/tex]
Therefore, the radius [tex]\(r\)[/tex] is:
[tex]\[
r = \frac{2\sqrt{5}}{2} = \sqrt{5}
\][/tex]
3. Form 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 [tex]\(h = 0\)[/tex], [tex]\(k = 0\)[/tex], and [tex]\(r = \sqrt{5}\)[/tex]:
[tex]\[
(x - 0)^2 + (y - 0)^2 = (\sqrt{5})^2
\][/tex]
4. Simplify the equation:
[tex]\[
x^2 + y^2 = 5
\][/tex]
The equation of the circle is:
[tex]\[
(x - 0)^2 + (y - 0)^2 = 5
\][/tex]
