Given:
- P1: parabola with vertex (3, 2) and focus (4, 4)
- P2: mirror image of P1 with respect to the line x - 2y = 6
- To find: the directrix of P2 in the form of x - 2y = ___

Step-by-step solution:
1. Determine the equation of P1:
- Since the vertex is (3, 2) and the focus is (4, 4), the axis of symmetry is parallel to the y-axis and the directrix is a horizontal line.
- Using the formula for the distance between a point (x, y) and a line Ax + By + C = 0, we can find the equation of the directrix:
distance[(x, y), x = 3] = distance[(x, y), y = 0]
(x - 3)^2 = (y - 0)^2 + (4 - 2)^2
(x - 3)^2 = y^2 + 4
This is the standard form of a parabola with vertex (3, 2), so P1 is:
(x - 3)^2 = 4(y - 2)

2. Find the image of P1 with respect to the line x - 2y = 6:
- The line x - 2y = 6 passes through the vertex (3, 2) of P1, so it is the axis of symmetry for P2.
- To find the image of P1, we reflect it across the axis of symmetry by replacing y with (x - 6)/(-2) (the equation of the line in slope-intercept form) in the equation of P1:
(x - 3)^2 = 4[(x - 6)/(-2) - 2]
(x - 3)^2 = -(x - 10)

3. Determine the equation of the directrix of P2:
- Since the vertex of P2 is on the line x - 2y = 6, the directrix is perpendicular to the line and passes through the vertex.
- The slope of x - 2y = 6 is 1/2, so the slope of the directrix is -2 (the negative reciprocal).
- Using the point-slope form of a line, we can find the equation of the directrix:
y - 2 = -2(x - 3)
y = -2x + 8
Rearranging, we get:
x - 2y = 8
- To get the answer in the desired form of x - 2y = ___, we subtract 6 from both sides:
x - 2y = 8 - 6
x - 2y = 2
Therefore, the directrix of P2 is x - 2y = 10.