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Let the lines $$\frac{x-1}{\lambda} = \frac{y-2}{1} = \frac{z-3}{2}$$ and $$\frac{x+26}{-2} = \frac{y+18}{3} = \frac{z+28}{\lambda}$$ be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lie on P?
We have lines $$L_1: \frac{x-1}{\lambda} = \frac{y-2}{1} = \frac{z-3}{2}$$ and $$L_2: \frac{x+26}{-2} = \frac{y+18}{3} = \frac{z+28}{\lambda}$$.
For the lines to be coplanar, the determinant formed by the direction vectors and the vector joining the points on the two lines must be zero.
The point on $$L_1$$ is $$(1, 2, 3)$$ and on $$L_2$$ is $$(-26, -18, -28)$$. The vector joining them is $$(-27, -20, -31)$$.
The direction vectors are $$(\lambda, 1, 2)$$ and $$(-2, 3, \lambda)$$.
The coplanarity condition is:
$$\begin{vmatrix} -27 & -20 & -31 \\ \lambda & 1 & 2 \\ -2 & 3 & \lambda \end{vmatrix} = 0$$
Expanding along the first row:
$$-27(\lambda - 6) + 20(\lambda^2 + 4) - 31(3\lambda + 2) = 0$$
$$-27\lambda + 162 + 20\lambda^2 + 80 - 93\lambda - 62 = 0$$
$$20\lambda^2 - 120\lambda + 180 = 0$$
$$\lambda^2 - 6\lambda + 9 = 0$$
$$(\lambda - 3)^2 = 0$$
So $$\lambda = 3$$.
Now we find the equation of the plane $$P$$ containing both lines. With $$\lambda = 3$$, the direction vectors are $$(3, 1, 2)$$ and $$(-2, 3, 3)$$.
The normal to the plane is their cross product:
$$\vec{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 1 & 2 \\ -2 & 3 & 3 \end{vmatrix} = \hat{i}(3-6) - \hat{j}(9+4) + \hat{k}(9+2) = (-3, -13, 11)$$
The plane passes through $$(1, 2, 3)$$:
$$-3(x-1) - 13(y-2) + 11(z-3) = 0$$
$$-3x + 3 - 13y + 26 + 11z - 33 = 0$$
$$-3x - 13y + 11z - 4 = 0$$
Or equivalently: $$3x + 13y - 11z + 4 = 0$$.
Now we check each point:
$$(0, -2, -2)$$: $$0 - 26 + 22 + 4 = 0$$ — lies on $$P$$.
$$(-5, 0, -1)$$: $$-15 + 0 + 11 + 4 = 0$$ — lies on $$P$$.
$$(3, -1, 0)$$: $$9 - 13 + 0 + 4 = 0$$ — lies on $$P$$.
$$(0, 4, 5)$$: $$0 + 52 - 55 + 4 = 1 \neq 0$$ — does NOT lie on $$P$$.
Hence, the correct answer is Option D.
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