Write a systems of equations, letting (x, y) be the centre. The logic is the following: since the two points given lie on the circumference of the circle, the distance to the centre is equal with both points, so let equation 1 be sqrt((x - 5)^2 + (y - 2)^2) = sqrt((x - 3)^2 + (y - 2)^2)
The centre lies on the line y = 2/3x + 7, so it makes sense that our second equation is y = 2/3x + 7.
Let's now solve using substitution.
sqrt((x - 5)^2 + (2/3x + 7 - 2)^2 ) = sqrt((x - 3)^2 + (2/3x + 7 - 2)^2)
x^2 - 10x + 25 + 4/9x^2 + 20/3x + 25 = x^2 - 6x + 9 + 4/9x^2 + 20/3x + 25
-4x = -16
x = 4
y = 2/3(4) + 7 = 8/3 + 7 = 29/3
Hence, the centre has coordinates of (4, 29/3).
Now, we need to determine the measure of the radius, again using the distance formula.
r = sqrt((5 - 4)^2 + (29/3 - 2)^2)
r = sqrt(1 + 529/9)
r = sqrt(538/9)
The most commonly used form of the equation of the circle is (x - a)^2 + (y - b)^2 = r^2, where (a, b) is the centre and r the radius.
Thus, the equation of the circle is (x - 4)^2 + (y - 29/3)^2 = 538/9
Hopefully this helps!
