A circle is tangent to the x axis at x=5:

the center is (5,k)

radius = k = root(5^2 + (k-3)^2) 

Square both sides: k^2 = 25 + (k^2 - 6k + 9)

So 6k = 34, hence k = 34/6 = 17/3

the center is (5,17/3)

the other y-intercept is 17/3 + (17/3 - 3) = 23/3

the equation of the circle is (x - 5)^2 + (y - 17/3)^2 = (17/3)^2