Final answer:
The center of the circle passing through points A(-7,3) and B(5,-13) is (-1,-5), and the radius is 10. Using these values, the equation of the circle in standard form is (x + 1)² + (y + 5)² = 100.
Step-by-step explanation:
To find the equation of the circle passing through the points A(-7,3) and B(5,-13), we need to determine the center and the radius of the circle. The center of the circle is the midpoint of the line segment AB, and we can find it by averaging the x-coordinates and the y-coordinates of A and B.
The center coordinates (Cx, Cy) are therefore calculated as follows:
- Cx = (x1 + x2) / 2 = (-7 + 5) / 2 = -1
- Cy = (y1 + y2) / 2 = (3 + (-13)) / 2 = -5
The radius can be found by calculating the distance between either A or B and the center. Using the distance formula, the radius r is:
r = √((x1 - Cx)² + (y1 - Cy)²)
Substituting the known values, we get:
r = √((-7 - (-1))² + (3 - (-5))²) = √((6)² + (8)²) = √(36 + 64) = √100 = 10
The standard form of the circle's equation is:
(x - Cx)² + (y - Cy)² = r²
Substituting the center coordinates and the radius, we obtain:
(x + 1)² + (y + 5)² = 10²
(x + 1)² + (y + 5)² = 100
This is the equation of the circle in standard form.