To find the equation of a circle given its diameter endpoints, we first need to determine the center and the radius of the circle. The endpoints of the diameter are the points (8, 2) and (2, 6).
Step 1: Calculate the center of the circle.
The center of the circle can be found by averaging the x-coordinates and the y-coordinates of the endpoints:
- Center x-coordinate: Cx = (8 + 2) / 2 = 10 / 2 = 5
- Center y-coordinate: Cy = (2 + 6) / 2 = 8 / 2 = 4
Thus, the center of the circle is located at (5, 4).
Step 2: Calculate the radius of the circle.
The radius can be determined by finding the distance from the center to one of the endpoints. We will use the distance formula:
Distance formula: d = √((x2 – x1)² + (y2 – y1)²)
Choosing the point (8, 2):
- r = √((8 – 5)² + (2 – 4)²)
- = √(3² + (-2)²)
- = √(9 + 4)
- = √13
So, the radius of the circle is √13.
Step 3: Write the equation of the circle.
The standard form of the equation of a circle is:
(x – Cx)² + (y – Cy)² = r²
Plugging in our values:
- Cx = 5
- Cy = 4
- r² = (√13)² = 13
Therefore, the equation of the circle is:
(x – 5)² + (y – 4)² = 13
This equation represents a circle centered at (5, 4) with a radius of √13.