1. Key Concepts and Formulas

• Section Formula (Internal Division): If a point C divides the line segment joining $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ internally in the ratio $m:n$, its coordinates are given by: $C = \left( \frac{nx_1 + mx_2}{m+n}, \frac{ny_1 + my_2}{m+n}, \frac{nz_1 + mz_2}{m+n} \right)$
• Condition for a point to lie on a plane: A point $(x_0, y_0, z_0)$ lies on the plane $Ax+By+Cz=D$ if and only if its coordinates satisfy the plane's equation: $Ax_0+By_0+Cz_0=D$.
• Distance Formula from Origin: The distance of a point $P(x,y,z)$ from the origin $O(0,0,0)$ is $OP = \sqrt{x^2+y^2+z^2}$.
• Distance Formula between two points: The square of the distance between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ is $P_1P_2^2 = (x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2$.

2. Step-by-Step Solution

Step 1: Determine the coordinates of point C using the section formula. We are given that point C divides the line segment joining A($a,-2,4$) and B($2,b,-3$) in the ratio 2:1. This means $m=2$ and $n=1$. Applying the section formula for internal division: $C = \left( \frac{1 \cdot a + 2 \cdot 2}{2+1}, \frac{1 \cdot (-2) + 2 \cdot b}{2+1}, \frac{1 \cdot 4 + 2 \cdot (-3)}{2+1} \right)$ $C = \left( \frac{a+4}{3}, \frac{-2+2b}{3}, \frac{4-6}{3} \right)$ $C = \left( \frac{a+4}{3}, \frac{2b-2}{3}, \frac{-2}{3} \right)$ These are the coordinates of point C in terms of $a$ and $b$.

Step 2: Utilize the condition that point C lies on the given plane. The plane is defined by the equation $2x-y+z=4$. Since point C lies on this plane, its coordinates must satisfy the plane's equation. Substitute the coordinates of C into the plane equation: $2\left( \frac{a+4}{3} \right) - \left( \frac{2b-2}{3} \right) + \left( \frac{-2}{3} \right) = 4$ To eliminate the denominators, multiply the entire equation by 3: $2(a+4) - (2b-2) - 2 = 12$ Expand and simplify the equation: $2a + 8 - 2b + 2 - 2 = 12$ $2a - 2b + 8 = 12$ $2a - 2b = 4$ Dividing by 2, we get a linear relationship between $a$ and $b$: $a - b = 2 \quad \text{(Equation 1)}$

Step 3: Use the distance of point C from the origin. We are given that the distance of point C from the origin $O(0,0,0)$ is $\sqrt{5}$. Therefore, $OC^2 = (\sqrt{5})^2 = 5$. Using the distance formula for C $\left( \frac{a+4}{3}, \frac{2b-2}{3}, \frac{-2}{3} \right)$ from the origin: $\left( \frac{a+4}{3} \right)^2 + \left( \frac{2b-2}{3} \right)^2 + \left( \frac{-2}{3} \right)^2 = 5$ $\frac{(a+4)^2}{9} + \frac{4(b-1)^2}{9} + \frac{4}{9} = 5$ Multiply the entire equation by 9 to clear the denominators: $(a+4)^2 + 4(b-1)^2 + 4 = 45$ $(a+4)^2 + 4(b-1)^2 = 41 \quad \text{(Equation 2)}$ Now we have a system of two equations with two variables. From Equation 1, we can express $a$ as $a = b+2$. Substitute this expression for $a$ into Equation 2: $((b+2)+4)^2 + 4(b-1)^2 = 41$ $(b+6)^2 + 4(b-1)^2 = 41$ Expand the squared terms: $(b^2 + 12b + 36) + 4(b^2 - 2b + 1) = 41$ $b^2 + 12b + 36 + 4b^2 - 8b + 4 = 41$ Combine like terms to form a quadratic equation in $b$: $5b^2 + 4b + 40 = 41$ $5b^2 + 4b - 1 = 0$ Factor the quadratic equation: $5b^2 + 5b - b - 1 = 0$ $5b(b+1) - 1(b+1) = 0$ $(5b-1)(b+1) = 0$ This yields two possible values for $b$: $b = \frac{1}{5} \quad \text{or} \quad b = -1$

Step 4: Determine the correct values for $a$ and $b$ using the condition $ab < 0$. We will evaluate each case using $a = b+2$:

• Case 1: If $b = \frac{1}{5}$ Then $a = \frac{1}{5} + 2 = \frac{1+10}{5} = \frac{11}{5}$. In this case, $ab = \left( \frac{11}{5} \right) \left( \frac{1}{5} \right) = \frac{11}{25}$. Since $\frac{11}{25} > 0$, this pair does not satisfy the condition $ab < 0$.

• Case 2: If $b = -1$ Then $a = -1 + 2 = 1$. In this case, $ab = (1)(-1) = -1$. Since $-1 < 0$, this pair satisfies the condition $ab < 0$.

Therefore, the correct values for $a$ and $b$ are $a=1$ and $b=-1$.

Step 5: Calculate the coordinates of points C and P. Using $a=1$ and $b=-1$:

• Coordinates of C: Substitute $a=1$ and $b=-1$ into the expression for C from Step 1: $C = \left( \frac{1+4}{3}, \frac{2(-1)-2}{3}, \frac{-2}{3} \right) = \left( \frac{5}{3}, \frac{-2-2}{3}, \frac{-2}{3} \right) = \left( \frac{5}{3}, \frac{-4}{3}, \frac{-2}{3} \right)$
• Coordinates of P: The point P is given as $(a-b, b, 2b-a)$. Substitute $a=1$ and $b=-1$: $P = (1 - (-1), -1, 2(-1) - 1)$ $P = (1 + 1, -1, -2 - 1)$ $P = (2, -1, -3)$

Step 6: Calculate $CP^2$. We need to find the square of the distance between $C \left( \frac{5}{3}, \frac{-4}{3}, \frac{-2}{3} \right)$ and $P (2, -1, -3)$. Using the distance formula: $CP^2 = \left( 2 - \frac{5}{3} \right)^2 + \left( -1 - \left( \frac{-4}{3} \right) \right)^2 + \left( -3 - \left( \frac{-2}{3} \right) \right)^2$ $CP^2 = \left( \frac{6-5}{3} \right)^2 + \left( \frac{-3+4}{3} \right)^2 + \left( \frac{-9+2}{3} \right)^2$ $CP^2 = \left( \frac{1}{3} \right)^2 + \left( \frac{1}{3} \right)^2 + \left( \frac{-7}{3} \right)^2$ $CP^2 = \frac{1}{9} + \frac{1}{9} + \frac{49}{9}$ $CP^2 = \frac{1+1+49}{9} = \frac{51}{9}$ Simplifying the fraction: $CP^2 = \frac{17}{3}$

3. Common Mistakes & Tips

• Ratio Order in Section Formula: Be careful with the order of $m$ and $n$ in the section formula. If C divides AB in ratio $m:n$, it's $nx_1+mx_2$.
• Algebraic Precision: Errors often occur in expanding squares, combining like terms, or solving quadratic equations. Double-check calculations, especially when dealing with fractions.
• Checking Conditions: Do not forget to use all given conditions, like $ab < 0$, to eliminate incorrect solutions for variables.
• Sign Errors: Pay close attention to negative signs during substitution and arithmetic operations.

4. Summary

The problem involved a sequence of steps starting with finding the coordinates of point C using the section formula. Then, we used the fact that C lies on the given plane and its distance from the origin to establish a system of two equations in $a$ and $b$. Solving this system, and applying the condition $ab<0$, allowed us to uniquely determine the values of $a$ and $b$. Finally, we calculated the coordinates of C and P using these values and then found the square of the distance between C and P.

5. Final Answer

The value of $CP^2$ is $\frac{17}{3}$.

The final answer is $\boxed{\frac{17}{3}}$ which corresponds to option (A).