The plane $$2x - y + z = 4$$ intersects the line segment joining the points $$A(a, -2, 4)$$ and $$B(2, b, -3)$$ at the point $$C$$ in the ratio $$2 : 1$$ and the distance of the point $$C$$ from the origin is $$\sqrt{5}$$. If $$ab < 0$$ and $$P$$ is the point $$(a-b, b, 2b-a)$$ then $$CP^2$$ is equal to:

Consider the plane $$2x - y + z = 4$$ and the points $$A(a, -2, 4)$$ and $$B(2, b, -3)$$. The point $$C$$ divides the segment $$AB$$ internally in the ratio $$2:1$$.

The coordinates of $$C$$ are obtained by internal division in the ratio $$2:1$$: $$C = \left(\frac{2\cdot 2 + 1\cdot a}{3}, \frac{2\cdot b + 1\cdot(-2)}{3}, \frac{2\cdot(-3) + 1\cdot 4}{3}\right) = \left(\frac{4 + a}{3}, \frac{2b - 2}{3}, -\frac{2}{3}\right).$$

Since $$C$$ lies on the plane $$2x - y + z = 4$$, substitute its coordinates to get $$2\cdot\frac{4 + a}{3} - \frac{2b - 2}{3} + \left(-\frac{2}{3}\right) = 4.$$ Clearing denominators: $$8 + 2a - 2b + 2 - 2 = 12 \implies 8 + 2a - 2b = 12 \implies a - b = 2.$$

The condition $$|OC| = \sqrt{5}$$ implies $$|OC|^2 = 5$$. Computing the square of the distance from the origin to $$C$$ gives $$\frac{(4 + a)^2 + (2b - 2)^2 + (-2)^2}{9} = 5 \quad\Longrightarrow\quad (4 + a)^2 + (2b - 2)^2 = 41.$$

Using $$a = b + 2$$ from above and substituting into the last equation yields $$(6 + b)^2 + (2b - 2)^2 = 41,$$ which expands to $$36 + 12b + b^2 + 4b^2 - 8b + 4 = 41 \implies 5b^2 + 4b - 1 = 0 \implies b = \frac{-4 \pm 6}{10}.$$ Hence, $$b = \frac{1}{5}$$ or $$b = -1$$.

Since the product $$ab$$ must be negative, we select $$b = -1$$, giving $$a = 1$$ and $$ab = -1 < 0$$. The other value yields $$ab > 0$$.

Thus $$a = 1$$ and $$b = -1$$. The corresponding point is $$C = \left(\frac{5}{3}, -\frac{4}{3}, -\frac{2}{3}\right),$$ and the point $$P$$, defined by $$(a - b,\, b,\, 2b - a),$$ becomes $$P = (2, -1, -3).$$

Finally, the squared distance $$CP^2$$ is $$\left(2 - \frac{5}{3}\right)^2 + \left(-1 + \frac{4}{3}\right)^2 + \left(-3 + \frac{2}{3}\right)^2 = \left(\frac{1}{3}\right)^2 + \left(\frac{1}{3}\right)^2 + \left(-\frac{7}{3}\right)^2 = \frac{1 + 1 + 49}{9} = \frac{51}{9} = \frac{17}{3}.$$ The correct answer is Option A.