If $$P(h, k)$$ be point on the parabola $$x = 4y^2$$, which is nearest to the point $$Q(0, 33)$$, then the distance of $$P$$ from the directrix of the parabola $$y^2 = 4(x + y)$$ is equal to:

Step 1: Find the coordinates of the point P on the first parabola

The equation of the first parabola is:

$$x = 4y^2$$

Let any parametric point on this parabola be $$P(4t^2, t)$$. We need to find the value of $$t$$ such that the distance from $$P$$ to the point $$Q(0, 33)$$ is minimized. The square of the distance $$D^2$$ between $$P$$ and $$Q$$ is given by:

$$D^2 = (4t^2 - 0)^2 + (t - 33)^2$$

$$D^2 = 16t^4 + t^2 - 66t + 1089$$

To minimize $$D^2$$, we differentiate it with respect to $$t$$ and set it to zero:

$$\frac{d(D^2)}{dt} = 64t^3 + 2t - 66 = 0$$

Dividing the entire equation by 2:

$$32t^3 + t - 33 = 0$$

By inspection, we can see that $$t = 1$$ is a real root of this cubic equation because $$32(1)^3 + 1 - 33 = 0$$. Since the function is strictly increasing, $$t = 1$$ is the only real solution.

Substituting $$t = 1$$ back into the parametric coordinates of $$P$$:

$$h = 4(1)^2 = 4$$

$$k = 1$$

Thus, the point closest to $$Q$$ is $$P(4, 1)$$.

Step 2: Find the equation of the directrix of the second parabola

The equation of the second parabola is given by:

$$y^2 = 4(x + y)$$

$$y^2 - 4y = 4x$$

Completing the square on the left-hand side:

$$(y - 2)^2 - 4 = 4x$$

$$(y - 2)^2 = 4x + 4$$

$$(y - 2)^2 = 4(x + 1)$$

Comparing this with the standard equation of a parabola $$(Y)^2 = 4aX$$, we identify:

- $$Y = y - 2$$

- $$X = x + 1$$

- $$4a = 4 \implies a = 1$$

The standard equation of the directrix for $$Y^2 = 4aX$$ is given by $$X = -a$$. Substituting our variables:

$$x + 1 = -1$$

$$x = -2$$

$$x + 2 = 0$$

Step 3: Calculate the distance of P from the directrix

We need to find the perpendicular distance of the point $$P(4, 1)$$ from the line $$x + 2 = 0$$:

$$\text{Distance} = \frac{|4 + 2|}{\sqrt{1^2 + 0^2}}$$

$$\text{Distance} = |6| = 6$$

Conclusion:

The distance of $$P$$ from the directrix of the parabola is equal to 6.