# A circle has a center that falls on the line #y = 3/8x +8 # and passes through # ( 7 ,4 )# and #(2 ,9 )#. What is the equation of the circle?

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The equation of the circle can be found using the formula for the standard form of a circle: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

First, we need to find the center of the circle. Since the center falls on the line y = (3/8)x + 8, we can set this equation equal to y and solve for x: y = (3/8)x + 8 Substitute y with 4 (from the point (7, 4)): 4 = (3/8)(7) + 8 Then, solve for x: x = 8

So, the x-coordinate of the center of the circle is 8.

Now, let's find the y-coordinate of the center by substituting x with 8 in the equation of the line: y = (3/8)(8) + 8 Simplify: y = 3 + 8 y = 11

So, the y-coordinate of the center of the circle is 11.

Now, we need to find the radius of the circle. We can use the distance formula between the center of the circle (h, k) and one of the given points on the circle to find the radius.

Using the point (7, 4): r^2 = (x2 - x1)^2 + (y2 - y1)^2 r^2 = (7 - 8)^2 + (4 - 11)^2 r^2 = (-1)^2 + (-7)^2 r^2 = 1 + 49 r^2 = 50

Now we have the center (h, k) = (8, 11) and the radius squared r^2 = 50. We can plug these values into the standard form of the circle equation: (x - 8)^2 + (y - 11)^2 = 50

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