# A circle has a center that falls on the line #y = 2/3x +7 # and passes through # ( 3 ,1 )# and #(6 ,4 )#. What is the equation of the circle?

The equation of the circle is

Therefore, the circle's equation is

graph{y-(2/3)x-7)(x+y-7)=0 [-17.5, 22.5, -4.24, 15.76]}

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The center of the circle can be found by finding the intersection point of the line y = (2/3)x + 7 and the perpendicular bisector of the line segment joining the given points (3, 1) and (6, 4).

First, find the slope of the line perpendicular to y = (2/3)x + 7: Perpendicular slope = -1/(2/3) = -3/2

Now, find the midpoint of the line segment joining (3, 1) and (6, 4): Midpoint = ((3 + 6)/2, (1 + 4)/2) = (4.5, 2.5)

Now, using the midpoint and the slope found, find the equation of the perpendicular bisector: Using point-slope form: y - y1 = m(x - x1) y - 2.5 = (-3/2)(x - 4.5) y - 2.5 = (-3/2)x + (3/2) * 4.5 y - 2.5 = (-3/2)x + 6.75 y = (-3/2)x + 9.25

Now, solve the system of equations to find the intersection point of this line with y = (2/3)x + 7: (2/3)x + 7 = (-3/2)x + 9.25 (2/3 + 3/2)x = 9.25 - 7 (4/6 + 9/6)x = 2.25 (13/6)x = 2.25 x = (2.25 * 6)/13 x ≈ 1.0385

Now, find y by substituting x back into either of the equations: y = (2/3)(1.0385) + 7 ≈ 7.6923

So, the center of the circle is approximately (1.0385, 7.6923).

Now, find the radius of the circle using one of the given points and the center: Radius = √((x2 - x1)^2 + (y2 - y1)^2) Radius = √((3 - 1.0385)^2 + (1 - 7.6923)^2) Radius ≈ √((1.9615)^2 + (6.6923)^2) Radius ≈ √(3.8486 + 44.7686) Radius ≈ √48.6172 Radius ≈ 6.9773

Therefore, the equation of the circle is (x - 1.0385)^2 + (y - 7.6923)^2 = (6.9773)^2.

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