To find the equation of the circle with [tex]\( P = (3, 1) \)[/tex] and [tex]\( Q = (-3, -7) \)[/tex] as the endpoints of its diameter, follow these steps:
1. Find the midpoint of segment PQ:
The midpoint of the line segment PQ will serve as the center of the circle. The coordinates of the midpoint can be calculated as follows:
[tex]\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Substituting the given coordinates [tex]\( P = (3, 1) \)[/tex] and [tex]\( Q = (-3, -7) \)[/tex]:
[tex]\[
\text{Midpoint} = \left( \frac{3 + (-3)}{2}, \frac{1 + (-7)}{2} \right) = (0, -3)
\][/tex]
So, the center of the circle is [tex]\( (0, -3) \)[/tex].
2. Calculate the length of the diameter:
The length of the segment PQ (which is the diameter of the circle) can be found using the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting the coordinates of [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(-3 - 3)^2 + (-7 - 1)^2} = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\][/tex]
Thus, the diameter of the circle is 10.
3. Find the radius of the circle:
The radius is half the length of the diameter:
[tex]\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{10}{2} = 5
\][/tex]
4. Write the equation of the circle:
The standard form of the equation of a circle is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
Here, the center [tex]\( (h, k) \)[/tex] is [tex]\( (0, -3) \)[/tex] and the radius [tex]\( r \)[/tex] is 5.
Substituting these values into the standard circle equation form gives:
[tex]\[
(x - 0)^2 + (y - (-3))^2 = 5^2
\][/tex]
This simplifies to:
[tex]\[
(x - 0)^2 + (y + 3)^2 = 25
\][/tex]
Therefore, the equation of the circle is:
[tex]\[
(x - 0)^2 + (y + 3)^2 = 25
\][/tex]
This is the required equation of the circle having segment PQ as its diameter.
