Final answer:
To find the equation of the circle with given diameter endpoints, we calculate the midpoint to get the center of the circle and then determine the radius. We then plug these values into the standard circle equation to get the circle's equation: (x - 3)^2 + (y + 0.5)^2 = 3.24 (approximately).
Step-by-step explanation:
To find the equation of the circle given the extremities of the diameter, first we need to determine the center of the circle, which is the midpoint of the diameter. The midpoint (h, k) can be found using the midpoint formula:
- h = (x1 + x2) / 2
- k = (y1 + y2) / 2
For the points (4, -2) and (2, 1), the center (h, k) is:
- h = (4 + 2) / 2 = 3
- k = (-2 + 1) / 2 = -0.5
Next, we calculate the radius, r, of the circle, which is the distance between the center and one of the endpoints of the diameter:
r = √ [(x1 - h)² + (y1 - k)² ]
Using point (4, -2) and center (3, -0.5), the radius, r, is:
r = √ [(4 - 3)² + (-2 + 0.5)² ] = √ [1 + 2.25] = √ 3.25 = 1.8 (approx.)
The equation of a circle with center (h, k) and radius r is:
(x - h)² + (y - k)² = r²
Substituting the center (3, -0.5) and radius 1.8 (approx.) into this formula gives us:
(x - 3)² + (y + 0.5)² = (1.8)²
(x - 3)² + (y + 0.5)² = 3.24 (approximately)