To find the equation of a circle given the endpoints [tex]\( P = (-1, -1) \)[/tex] and [tex]\( Q = (5, 3) \)[/tex] of its diameter, we can follow these steps:
1. Calculate the center of the circle:
- The center of the circle is the midpoint of the diameter, which can be found using the midpoint formula.
[tex]\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
- For the points [tex]\( P = (-1, -1) \)[/tex] and [tex]\( Q = (5, 3) \)[/tex]:
[tex]\[
\text{Center} = \left( \frac{-1 + 5}{2}, \frac{-1 + 3}{2} \right) = (2.0, 1.0)
\][/tex]
So, the center of the circle is at [tex]\( (2.0, 1.0) \)[/tex].
2. Calculate the radius of the circle:
- The diameter length can be found using the distance formula between [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
- For the points [tex]\( P = (-1, -1) \)[/tex] and [tex]\( Q = (5, 3) \)[/tex]:
[tex]\[
\text{Diameter} = \sqrt{(5 - (-1))^2 + (3 - (-1))^2} = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} = 7.211102550927978
\][/tex]
- The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
\text{Radius} = \frac{7.211102550927978}{2} = 3.605551275463989
\][/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]
- Substituting [tex]\( h = 2.0 \)[/tex], [tex]\( k = 1.0 \)[/tex], and [tex]\( r = 3.605551275463989 \)[/tex]:
[tex]\[
(x - 2.0)^2 + (y - 1.0)^2 = (3.605551275463989)^2
\][/tex]
- Finally, we compute [tex]\( r^2 \)[/tex]:
[tex]\[
(3.605551275463989)^2 = 12.999999999999998
\][/tex]
Therefore, the equation of the circle is:
[tex]\[
(x - 2.0)^2 + (y - 1.0)^2 = 12.999999999999998
\][/tex]
