To find the equation of a circle when 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. If [tex]\( P = (-2, -1) \)[/tex] and [tex]\( Q = (2, 1) \)[/tex] are the endpoints of the diameter, the midpoint [tex]\( M \)[/tex] is calculated as follows:
[tex]\[
\text{Midpoint} = \left( \frac{-2 + 2}{2}, \frac{-1 + 1}{2} \right) = (0, 0)
\][/tex]
So, the center of the circle is at [tex]\( (0, 0) \)[/tex].
2. Calculate the radius of the circle:
The radius is half the length of the diameter. To find this, calculate the distance between [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
[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]
Since this distance represents the diameter, the radius [tex]\( r \)[/tex] is:
[tex]\[
r = \frac{2\sqrt{5}}{2} = \sqrt{5}
\][/tex]
3. Write the equation of the circle:
The general equation for 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 center [tex]\( (0, 0) \)[/tex] and radius [tex]\( \sqrt{5} \)[/tex] into the equation, we obtain:
[tex]\[
(x - 0)^2 + (y - 0)^2 = (\sqrt{5})^2
\][/tex]
Simplify this to get the final equation:
[tex]\[
x^2 + y^2 = 5
\][/tex]
Therefore, the equation of the circle is:
[tex]\[
(x - 0)^2 + (y - 0)^2 = 5
\][/tex]
[tex]\[
(x - 0)^2 + (y - 0)^2 = 5
\][/tex]
