Final answer:
To find the equation of the circle for which (-2,1) and (10,17) are endpoints of a diameter, we first find the center of the circle using the midpoint formula. Then, we use the distance formula to find the radius. Finally, we plug in the values of the center and radius into the equation of a circle.
Step-by-step explanation:
To find the equation of a circle for which the points (-2,1) and (10,17) are endpoints of the diameter, we first need to find the center of the circle. The center of a circle is the midpoint of its diameter, so we can use the midpoint formula: (x,y) = ((x1 + x2)/2, (y1 + y2)/2). Plugging in the coordinates of the two endpoints, we get:
(x,y) = ((-2 + 10)/2, (1 + 17)/2) = (4,9)
Now that we have the center of the circle, we can use the equation of a circle: (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center and r is the radius. Since the two given points are on the diameter, the distance between them is equal to the diameter, which is twice the radius:
Distance between points = 2r
Using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Plugging in the coordinates of the two endpoints:
Distance = sqrt((10 - (-2))^2 + (17 - 1)^2) = sqrt(12^2 + 16^2) = sqrt(144 + 256) = sqrt(400) = 20
Since the distance between the two points is equal to 2r, we have:
20 = 2r
Simplifying, we find that r = 10. Now we can plug in the values of the center and the radius into the equation of a circle:
(x - 4)^2 + (y - 9)^2 = 10^2