To find the equation of the circle that has segment PQ as its diameter, we need to determine the center and the radius of the circle.
Step 1: Find the coordinates of the midpoint of segment PQ
The midpoint of a line segment with endpoints [tex]\( P (x_1, y_1) \)[/tex] and [tex]\( Q (x_2, y_2) \)[/tex] is given by:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
For points [tex]\(P = (3, 1)\)[/tex] and [tex]\(Q = (-3, -7)\)[/tex], the coordinates of the midpoint are:
[tex]\[
\left( \frac{3 + (-3)}{2}, \frac{1 + (-7)}{2} \right) = \left( \frac{0}{2}, \frac{-6}{2} \right) = (0, -3)
\][/tex]
Thus, the center of the circle is at [tex]\( (0, -3) \)[/tex].
Step 2: Calculate the distance between points P and Q to determine the diameter
The distance between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting the coordinates of [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
[tex]\[
d = \sqrt{(-3 - 3)^2 + (-7 - 1)^2} = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\][/tex]
So, the diameter of the circle is 10 units.
Step 3: Determine the radius of the circle
The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
r = \frac{d}{2} = \frac{10}{2} = 5
\][/tex]
Step 4: Write the equation of the circle
The general 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]
For the center at [tex]\( (0, -3) \)[/tex] and radius [tex]\( 5 \)[/tex]:
[tex]\[
(x - 0)^2 + (y - (-3))^2 = 5^2
\][/tex]
Simplifying, the equation of the circle is:
[tex]\[
x^2 + (y + 3)^2 = 25
\][/tex]
Thus, the equation of the circle is:
[tex]\[
x^2 + (y + 3)^2 = 25
\][/tex]
