Certainly! To find the equation of the circle that has segment [tex]\( PQ \)[/tex] as its diameter, we need to follow these steps:
### 1. Find the midpoint of the segment PQ:
The midpoint of a segment is given by the average of the x-coordinates and the average of the y-coordinates of the endpoints.
Given points [tex]\( P = (-3, 5) \)[/tex] and [tex]\( Q = (1, 9) \)[/tex], the midpoint, which will be the center of the circle, is calculated as:
[tex]\[ \text{Midpoint} = \left( \frac{-3 + 1}{2}, \frac{5 + 9}{2} \right) \][/tex]
[tex]\[ \text{Midpoint} = (-1.0, 7.0) \][/tex]
### 2. Find the radius of the circle:
The radius of the circle is half the length of segment [tex]\( PQ \)[/tex]. The length of [tex]\( PQ \)[/tex] can be found using the distance formula:
[tex]\[ \text{Distance} = \sqrt{(1 - (-3))^2 + (9 - 5)^2} \][/tex]
[tex]\[ \text{Distance} = \sqrt{(4)^2 + (4)^2} \][/tex]
[tex]\[ \text{Distance} = \sqrt{16 + 16} \][/tex]
[tex]\[ \text{Distance} = \sqrt{32} \][/tex]
[tex]\[ \text{Radius} = \frac{\sqrt{32}}{2} = \sqrt{8} \][/tex]
### 3. 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]
Given:
- The center [tex]\((h, k) = (-1, 7)\)[/tex]
- The radius squared [tex]\( r^2 = 8 \)[/tex]
The equation of the circle becomes:
[tex]\[ (x + 1)^2 + (y - 7)^2 = 8 \][/tex]
Hence, the equation of the circle that has segment [tex]\( PQ \)[/tex] as its diameter is:
[tex]\[ (x + 1)^2 + (y - 7)^2 = 8.000000000000002 \][/tex]
