Let the image of the point $$P(0, -5, 0)$$ in the line $$\dfrac{x - 1}{2} = \dfrac{y}{1} = \dfrac{z + 1}{-2}$$ be the point R and the image of the point $$Q\left(0, \dfrac{-1}{2}, 0\right)$$ in the line $$\dfrac{x - 1}{-1} = \dfrac{y + 9}{4} = \dfrac{z + 1}{1}$$ be the point S. Then the square of the area of the parallelogram PQRS is __________.

$$\frac{x-1}{2} = \frac{y}{1} = \frac{z+1}{-2} = \lambda$$

$$M = (2\lambda + 1, \lambda, -2\lambda - 1)$$

$$\vec{PM} = (2\lambda + 1 - 0)\hat{i} + (\lambda - (-5))\hat{j} + (-2\lambda - 1 - 0)\hat{k}$$

$$\vec{PM} = (2\lambda + 1)\hat{i} + (\lambda + 5)\hat{j} + (-2\lambda - 1)\hat{k}$$

Since $$M$$ is the foot of the perpendicular from $$P$$, $$\vec{PM}$$ must be perpendicular to the line's direction vector $$\vec{d}_1 = 2\hat{i} + \hat{j} - 2\hat{k}$$:

$$\vec{PM} \cdot \vec{d}_1 = 0$$

$$2(2\lambda + 1) + 1(\lambda + 5) - 2(-2\lambda - 1) = 0$$

$$9\lambda + 9 = 0 \implies \lambda = -1$$

$$M = (2(-1) + 1, -1, -2(-1) - 1) = (-1, -1, 1)$$

Since $$M$$ is the midpoint of $$P(0, -5, 0)$$ and its image $$R(x_1, y_1, z_1)$$:

$$R = 2M - P = 2(-1, -1, 1) - (0, -5, 0) = (-2, 3, 2)$$

$$\frac{x-1}{-1} = \frac{y+9}{4} = \frac{z+1}{1} = \mu$$

$$N = (-\mu + 1, 4\mu - 9, \mu - 1)$$

$$\vec{QN} = (-\mu + 1 - 0)\hat{i} + \left(4\mu - 9 - \left(-\frac{1}{2}\right)\right)\hat{j} + (\mu - 1 - 0)\hat{k}$$

$$\vec{QN} = (-\mu + 1)\hat{i} + \left(4\mu - \frac{17}{2}\right)\hat{j} + (\mu - 1)\hat{k}$$

$$\vec{QN} \cdot \vec{d}_2 = 0$$

$$-1(-\mu + 1) + 4\left(4\mu - \frac{17}{2}\right) + 1(\mu - 1) = 0$$

$$18\mu - 36 = 0 \implies \mu = 2$$

$$N = (-2 + 1, 4(2) - 9, 2 - 1) = (-1, -1, 1)$$

$$S = 2N - Q = 2(-1, -1, 1) - \left(0, -\frac{1}{2}, 0\right) = \left(-2, -\frac{3}{2}, 2\right)$$

We have, $$P(0, -5, 0), \quad Q\left(0, -\frac{1}{2}, 0\right), \quad R(-2, 3, 2), \quad S\left(-2, -\frac{3}{2}, 2\right)$$

$$\vec{PQ} = (0 - 0)\hat{i} + \left(-\frac{1}{2} - (-5)\right)\hat{j} + (0 - 0)\hat{k} = \frac{9}{2}\hat{j}$$

$$\vec{PR} = (-2 - 0)\hat{i} + (3 - (-5))\hat{j} + (2 - 0)\hat{k} = -2\hat{i} + 8\hat{j} + 2\hat{k}$$

$$\vec{PQ} \times \vec{PR} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & \frac{9}{2} & 0 \\ -2 & 8 & 2 \end{vmatrix}$$

$$\vec{PQ} \times \vec{PR} = \hat{i}\left(\frac{9}{2} \times 2 - 0\right) - \hat{j}(0 - 0) + \hat{k}\left(0 - \left(-2 \times \frac{9}{2}\right)\right)$$

$$\vec{PQ} \times \vec{PR} = 9\hat{i} + 9\hat{k}$$

$$\text{Area}^2 = |\vec{PQ} \times \vec{PR}|^2 = 9^2 + 9^2 = 81 + 81 = 162$$