Answer: To solve this problem, we can use the fact that the center of a circle is equidistant from any point on the circle. Since the line x = 1 is tangent to the circle at (1, 6), the radius of the circle is perpendicular to this line, and passes through the point (1,6). Therefore, we can find the equation of the line that passes through (1,6) and is perpendicular to x=1, which is y = -1/2(x-1) + 6.
Next, we can use the fact that the center of the circle is on the line y=2x to find the point of intersection of this line and the line we just found. Solving the equations y=2x and y = -1/2(x-1) + 6, we get x = 2 and y = 4.
Therefore, the center of the circle is (2,4), and the radius is the distance from this center to the point (1,6), which is sqrt((2-1)^2 + (4-6)^2)=sqrt(2).
So, the center of the circle is (2,4) and the radius is sqrt(2).