Question

Find the standard form of the equation of the circle for which the endpoints are (-2, 4) and (8, 4). Show your work.

a) (x - 3)^2 + (y - 4)^2 = 5^2
b) (x + 3)^2 + (y - 4)^2 = 5^2
c) (x - 3)^2 + (y + 4)^2 = 5^2
d) (x + 3)^2 + (y + 4)^2 = 5^2

Answer 1

The standard form of the equation for the circle with endpoints (-2, 4) and (8, 4) is (x - 3)^2 + (y - 4)^2 = 5^2, which corresponds to option a.

Step-by-step explanation:

To find the standard form of the equation of a circle with endpoints (-2, 4) and (8, 4), we first determine the circle's center and radius. The center is the midpoint of the line segment with the given endpoints. We calculate it using the midpoint formula: \((x_1 + x_2)/2, (y_1 + y_2)/2) = ((-2 + 8)/2, (4 + 4)/2) = (3, 4)\). The radius is half the distance between the endpoints, which is the distance in the x-direction in this case because the y-coordinates are the same. Thus, the radius is \((8 - (-2))/2 = 5\).

The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). Substituting \(h = 3\), \(k = 4\), and \(r = 5\) into this formula gives us \((x - 3)^2 + (y - 4)^2 = 5^2\). Therefore, option a is the correct equation of the circle: \((x - 3)^2 + (y - 4)^2 = 5^2\).