Final answer:
To write the equation of a circle with given endpoints of the diameter, calculate the midpoint to find the center, obtain the radius using the distance formula, and then use the general form of the equation of a circle incorporating these values.
Step-by-step explanation:
The question is about finding the equation of a circle given the endpoints of its diameter. To find the equation of a circle, we must find the center and the radius of the circle. The center of the circle is the midpoint of the diameter, and the radius can be calculated by finding the distance between the center and one of the endpoints.
Step-by-step solution:
- Identify the endpoints of the diameter. In this case, the endpoints are (-4,6) and (30,18).
- Calculate the midpoint of the diameter, which will be the center of the circle. The midpoint formula is ((x1 + x2)/2, (y1 + y2)/2). So, the midpoint is ((-4 + 30)/2, (6 + 18)/2) = (13, 12).
- Calculate the radius of the circle by using the distance formula to find the distance between the midpoint and one of the endpoints. The distance formula is √((x2 - x1)² + (y2 - y1)²). Substituting the values in, we get the radius as √((30 - 13)² + (18 - 12)²) = √(289 + 36) = √(325).
- Write the equation of the circle using the general form (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Thus, the equation is (x - 13)² + (y - 12)² = 325.