A circle is defined as the set of all points that are equidistant from a fixed point called the center. The distance from any point on the circle to the center is called the radius. To define the equation of a circle, we need to know the center (h, k) and the radius r.
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The standard form of the equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
Utilizing the Diameter
In this problem, we are given the endpoints of a diameter, which is a line segment that passes through the center of the circle and has its endpoints on the circle. This information allows us to find both the center and the radius of the circle.
Finding the Center
The midpoint of the diameter is the center of the circle. To find the midpoint, we use the midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Let's apply this to our problem:
Given the endpoints (-6, 3) and (2, 1), we can calculate the midpoint as:
Midpoint = ((-6 + 2)/2, (3 + 1)/2) = (-2, 2)
Therefore, the center of the circle is (-2, 2).
Finding the Radius
The radius is half the length of the diameter. We can find the length of the diameter using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Applying this to our endpoints:
Distance = √((2 - (-6))^2 + (1 - 3)^2) = √(64 + 4) = √68 = 2√17
Therefore, the radius is half the length of the diameter, which is √17.
Writing the Equation
Now that we know the center (-2, 2) and the radius √17, we can substitute these values into the standard form of the equation of a circle:
(x - (-2))^2 + (y - 2)^2 = (√17)^2
Simplifying the equation, we get:
(x + 2)^2 + (y - 2)^2 = 17
Conclusion
By leveraging the properties of a circle and its diameter, we have successfully determined the equation of the circle given the endpoints of its diameter. The equation (x + 2)^2 + (y - 2)^2 = 17 represents the circle that passes through the points (-6, 3) and (2, 1) as the endpoints of its diameter.