Final answer:
The equations of the two circles tangent to the parabola y² = 4x at the point (1, 2) with a radius of 6 are (x - 7)² + (y + 4)² = 36 and (x + 5)² + (y - 8)² = 36.
Step-by-step explanation:
To find the equations of the two circles that are tangent to the parabola y² = 4x at the point (1, 2), we need to determine the position of the centers of these circles. Since the radius is given as 6, the centers will lie on the line normal to the parabola at the tangent point.
The slope of the parabola at any point is given by the derivative of y with respect to x. For y² = 4x, differentiating with respect to x gives us 2y(dy/dx) = 4, so at (1, 2), the slope is 1. The slope of the normal line is the negative reciprocal of the slope of the tangent line, which is -1. Therefore, the equation of the normal line at (1, 2) is y - 2 = -(x - 1) or x + y = 3.
The centers of the circles will be 6 units away from (1, 2) along this normal line. To find these points, we use the distance formula. If (h, k) is the center of one of the circles, then using the distance formula we get √[(h - 1)² + (k - 2)²] = 6. Substituting k = 3 - h from the normal line equation and solving, we find that there are two centers: (7, -4) and (-5, 8).
The equations of the two circles are (x - 7)² + (y + 4)² = 36 and (x + 5)² + (y - 8)² = 36, corresponding to the centers (7, -4) and (-5, 8) respectively.