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 steps:
1. Determine the center of the circle (the midpoint of [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]):
[tex]\[
\text{center}_x = \frac{-2 + 2}{2} = 0
\][/tex]
[tex]\[
\text{center}_y = \frac{-1 + 1}{2} = 0
\][/tex]
So, the center of the circle is [tex]\((0, 0)\)[/tex].
2. Calculate the radius of the circle (half the distance between [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]):
[tex]\[
\text{distance} = \sqrt{(2 - (-2))^2 + (1 - (-1))^2}
\][/tex]
[tex]\[
\text{distance} = \sqrt{(2 + 2)^2 + (1 + 1)^2}
\][/tex]
[tex]\[
\text{distance} = \sqrt{4^2 + 2^2}
\][/tex]
[tex]\[
\text{distance} = \sqrt{16 + 4} = \sqrt{20}
\][/tex]
[tex]\[
\text{radius} = \frac{\sqrt{20}}{2} = \frac{2\sqrt{5}}{2} = \sqrt{5}
\][/tex]
3. Form 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]
Substituting the values we found:
[tex]\[
(x - 0)^2 + (y - 0)^2 = (\sqrt{5})^2
\][/tex]
[tex]\[
x^2 + y^2 = 5
\][/tex]
Thus, the equation of the circle is
[tex]\[
(x - 0)^2 + (y - 0)^2 = 5 \implies x^2 + y^2 = 5
\][/tex]
