Let $$P_1$$ be a parabola with vertex $$(3, 2)$$ and focus $$(4, 4)$$ and $$P_2$$ be its mirror image with respect to the line $$x + 2y = 6$$. Then the directrix of $$P_2$$ is $$x + 2y =$$ ______.

Parabola $$P_1$$ has vertex $$V = (3, 2)$$ and focus $$F = (4, 4)$$.

The axis direction of $$P_1$$ is along $$\vec{VF} = (1, 2)$$.

The distance from vertex to focus: $$a = \sqrt{1^2 + 2^2} = \sqrt{5}$$.

The directrix of $$P_1$$ is the line perpendicular to the axis, passing through the point obtained by going distance $$a$$ from the vertex in the opposite direction of the focus.

The point on the directrix side: $$D = V - \frac{a \cdot \vec{VF}}{|\vec{VF}|} = (3, 2) - \frac{\sqrt{5} \cdot (1,2)}{\sqrt{5}} = (3-1, 2-2) = (2, 0)$$

The directrix of $$P_1$$ passes through $$(2, 0)$$ and is perpendicular to $$(1, 2)$$:

$$1(x - 2) + 2(y - 0) = 0 \implies x + 2y = 2$$

Now we reflect $$P_1$$ about the line $$x + 2y = 6$$.

The mirror line is $$x + 2y = 6$$ and the directrix of $$P_1$$ is $$x + 2y = 2$$.

Since both lines have the same normal direction $$(1, 2)$$, they are parallel.

The distance from the directrix $$x + 2y = 2$$ to the mirror line $$x + 2y = 6$$ is $$\frac{|6-2|}{\sqrt{5}} = \frac{4}{\sqrt{5}}$$.

When reflecting a line parallel to the mirror across the mirror, the reflected line is at the same distance on the other side.

The directrix of $$P_2$$ is: $$x + 2y = 6 + (6 - 2) = 10$$

The correct answer is $$10$$.