To find the equation of the circle that has a segment [tex]\( PQ \)[/tex] as its diameter, let's go through the steps systematically:
1. Identify Points [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
- Point [tex]\( P \)[/tex] is given as [tex]\( (3, 1) \)[/tex].
- Point [tex]\( Q \)[/tex] is given as [tex]\( (-3, -7) \)[/tex].
2. Find the Midpoint of [tex]\( PQ \)[/tex]:
The midpoint of segment [tex]\( PQ \)[/tex] will be the center of the circle. The midpoint [tex]\( (C_x, C_y) \)[/tex] can be calculated using the midpoint formula:
[tex]\[
C_x = \frac{x_1 + x_2}{2}, \qquad C_y = \frac{y_1 + y_2}{2}
\][/tex]
Here, [tex]\( (x_1, y_1) = (3, 1) \)[/tex] and [tex]\( (x_2, y_2) = (-3, -7) \)[/tex].
[tex]\[
C_x = \frac{3 + (-3)}{2} = \frac{0}{2} = 0
\][/tex]
[tex]\[
C_y = \frac{1 + (-7)}{2} = \frac{-6}{2} = -3
\][/tex]
Therefore, the center of the circle is at [tex]\( (0, -3) \)[/tex].
3. Calculate the Radius of the Circle:
The radius is half the length of segment [tex]\( PQ \)[/tex]. First, calculate the distance between points [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
[tex]\[
\text{Distance} = \sqrt{(-3 - 3)^2 + (-7 - 1)^2} = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\][/tex]
Since the diameter of the circle is 10, the radius [tex]\( r \)[/tex] is:
[tex]\[
r = \frac{10}{2} = 5
\][/tex]
4. Form the Equation of the Circle:
The standard form of a circle's equation is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\( (h, k) \)[/tex] is the center and [tex]\( r \)[/tex] is the radius. substituting [tex]\( h = 0 \)[/tex], [tex]\( k = -3 \)[/tex], and [tex]\( r = 5 \)[/tex]:
[tex]\[
(x - 0)^2 + (y - (-3))^2 = 5^2
\][/tex]
Simplifying this:
[tex]\[
x^2 + (y + 3)^2 = 25
\][/tex]
Thus, the equation of the circle that has the segment [tex]\( PQ \)[/tex] as its diameter is:
[tex]\[
(x - 0)^2 + (y - (-3))^2 = 25
\][/tex]
or more simply:
[tex]\[
(x - 0)^2 + (y + 3)^2 = 25
\][/tex]
