Final answer:
To find the equations of the two circles that are tangent to the graph of y^2 = 4x at the point (1, 2), we need to determine their centers. By using the distance formula and solving the equation, we find two possible values for the y-coordinate of the centers. The equations of the circles are then formed using the center coordinates and the radius of 6.
Step-by-step explanation:
To find the equations of the two circles, we need to determine their centers. The given equation of the graph is y^2 = 4x, which is in the standard form of a parabola. The vertex of the parabola is (0, 0). Since the two circles are tangent to the graph at the point (1, 2), the centers of the circles must lie on the line x = 1.
Let's denote the y-coordinate of the centers as h. Since the distance from the center to the point of tangency is equal to the radius of the circle (which is 6), we can use the distance formula to find the value of h:
Distance = sqrt((1 - x)^2 + (2 - y)^2)
6 = sqrt((1 - 1)^2 + (2 - h)^2)
6 = sqrt((2 - h)^2)
36 = (2 - h)^2
When we solve this equation, we get two possible values of h: h = -4 and h = 8.
Therefore, the equations of the two circles are:
(x - 1)^2 + (y + 4)^2 = 6^2
(x - 1)^2 + (y - 8)^2 = 6^2