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 `sqrt5` If ab`lt`0 and P is the point (a - b, b, 2b - a ) then `CP^2` is equal to

Options

A. `73/3`
B. `97/3`
C. `16/3`
D. `17/3`

Answer

To solve the problem step by step, we will follow the outlined process to find the coordinates of point C, the distance from the origin, and finally calculate \( CP^2 \).

### Step 1: Find the coordinates of point C
The coordinates of point C, which divides the line segment AB in the ratio 2:1, can be calculated using the section formula.

Given points:
- \( A(a, -2, 4) \)
- \( B(2, b, -3) \)

Using the section formula:
\[ C = \left( \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}, \frac{m_1 z_2 + m_2 z_1}{m_1 + m_2} \right) \]
where \( m_1 = 2 \) and \( m_2 = 1 \).

Calculating the coordinates of C:
- \( x_C = \frac{2 \cdot 2 + 1 \cdot a}{2 + 1} = \frac{4 + a}{3} \)
- \( y_C = \frac{2 \cdot b + 1 \cdot (-2)}{2 + 1} = \frac{2b - 2}{3} \)
- \( z_C = \frac{2 \cdot (-3) + 1 \cdot 4}{2 + 1} = \frac{-6 + 4}{3} = \frac{-2}{3} \)

Thus, the coordinates of point C are:
\[ C\left(\frac{4 + a}{3}, \frac{2b - 2}{3}, -\frac{2}{3}\right) \]

### Step 2: Check if point C lies on the plane
The plane equation is given as:
\[ 2x - y + z = 4 \]
Substituting the coordinates of C into the plane equation:
\[ 2\left(\frac{4 + a}{3}\right) - \left(\frac{2b - 2}{3}\right) + \left(-\frac{2}{3}\right) = 4 \]
Multiplying through by 3 to eliminate the denominator:
\[ 2(4 + a) - (2b - 2) - 2 = 12 \]
Expanding and simplifying:
\[ 8 + 2a - 2b + 2 - 2 = 12 \]
\[ 2a - 2b + 8 = 12 \]
\[ 2a - 2b = 4 \implies a - b = 2 \quad \text{(Equation 1)} \]

### Step 3: Find the distance of point C from the origin
The distance from the origin to point C is given as \( \sqrt{5} \):
\[ OC = \sqrt{\left(\frac{4 + a}{3}\right)^2 + \left(\frac{2b - 2}{3}\right)^2 + \left(-\frac{2}{3}\right)^2} = \sqrt{5} \]
Squaring both sides:
\[ \left(\frac{4 + a}{3}\right)^2 + \left(\frac{2b - 2}{3}\right)^2 + \left(-\frac{2}{3}\right)^2 = 5 \]
Multiplying by 9:
\[ (4 + a)^2 + (2b - 2)^2 + 4 = 45 \]
Simplifying:
\[ (4 + a)^2 + (2b - 2)^2 = 41 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations
From Equation 1, we have \( a = b + 2 \). Substitute \( a \) in Equation 2:
\[ (4 + (b + 2))^2 + (2b - 2)^2 = 41 \]
\[ (6 + b)^2 + (2b - 2)^2 = 41 \]
Expanding:
\[ (36 + 12b + b^2) + (4b^2 - 8b + 4) = 41 \]
Combining like terms:
\[ 5b^2 + 4b - 1 = 0 \]
Using the quadratic formula:
\[ b = \frac{-4 \pm \sqrt{16 + 20}}{10} = \frac{-4 \pm \sqrt{36}}{10} = \frac{-4 \pm 6}{10} \]
Thus, \( b = \frac{2}{10} = \frac{1}{5} \) or \( b = \frac{-10}{10} = -1 \).

### Step 5: Determine values of a and check condition \( ab < 0 \)
If \( b = \frac{1}{5} \), then \( a = \frac{1}{5} + 2 = \frac{11}{5} \) (both positive, not valid).
If \( b = -1 \), then \( a = -1 + 2 = 1 \) (valid since \( ab < 0 \)).

### Step 6: Find point P
Using \( a = 1 \) and \( b = -1 \):
\[ P = (a - b, b, 2b - a) = (1 - (-1), -1, 2(-1) - 1) = (2, -1, -3) \]

### Step 7: Calculate \( CP^2 \)
Coordinates of C:
\[ C\left(\frac{4 + 1}{3}, \frac{2(-1) - 2}{3}, -\frac{2}{3}\right) = \left(\frac{5}{3}, -\frac{4}{3}, -\frac{2}{3}\right) \]
Coordinates of P:
\[ P(2, -1, -3) \]
Calculating \( CP^2 \):
\[ CP^2 = \left(2 - \frac{5}{3}\right)^2 + \left(-1 + \frac{4}{3}\right)^2 + \left(-3 + \frac{2}{3}\right)^2 \]
Calculating each term:
1. \( \left(2 - \frac{5}{3}\right)^2 = \left(\frac{6}{3} - \frac{5}{3}\right)^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \)
2. \( \left(-1 + \frac{4}{3}\right)^2 = \left(-\frac{3}{3} + \frac{4}{3}\right)^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \)
3. \( \left(-3 + \frac{2}{3}\right)^2 = \left(-\frac{9}{3} + \frac{2}{3}\right)^2 = \left(-\frac{7}{3}\right)^2 = \frac{49}{9} \)

Adding them up:
\[ CP^2 = \frac{1}{9} + \frac{1}{9} + \frac{49}{9} = \frac{51}{9} = \frac{17}{3} \]

### Final Answer
Thus, \( CP^2 = \frac{17}{3} \).