Final answer:
To find the standard form of the circle's equation, we need to find the center and radius of the circle. The line segment through the center of the circle intersects the circle at the points (2,2) and (4,4). Using the midpoint formula and the distance formula, we can determine that the center of the circle is (3, 3) and the radius is sqrt(2). The standard form of the circle's equation is (x - 3)^2 + (y - 3)^2 = 2.
Step-by-step explanation:
To write the standard form of the circle's equation, we need to find the center and radius of the circle. The line segment through the center of the circle intersects the circle at the points (2,2) and (4,4). These points are on the line that passes through the center and have the same distance from the center, which is the radius of the circle. We can use the midpoint formula to find the center, which is the midpoint of the line segment, and the distance formula to find the radius of the circle.
1. Find the midpoint of the line segment using the midpoint formula:
(x,y) = ((x1 + x2)/2, (y1 + y2)/2)
(x,y) = ((2 + 4)/2, (2 + 4)/2)
(x,y) = (3, 3)
2. Find the distance between one of the points and the center using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((4 - 3)^2 + (4 - 3)^2)
d = sqrt(1 + 1)
d = sqrt(2)
Therefore, the center of the circle is (3, 3) and the radius is sqrt(2). The standard form of the circle's equation is (x - 3)^2 + (y - 3)^2 = 2.