Find the ratio in which the line segment joining the points (2, -1, 3) and (-1, 2, 1) is divided by the plane x + y + z = 5.

Answer

To find the ratio in which the line segment joining the points \( A(2, -1, 3) \) and \( B(-1, 2, 1) \) is divided by the plane \( x + y + z = 5 \), we can follow these steps:

### Step 1: Assume the Ratio
Let the ratio in which the line segment is divided be \( k:1 \). This means that the point \( C \) that divides the segment \( AB \) in the ratio \( k:1 \) can be expressed in terms of \( k \).

### Step 2: Find the Coordinates of Point C
Using the section formula in three dimensions, the coordinates of point \( C \) can be calculated as follows:
\[ C\left( \frac{-k + 2}{k + 1}, \frac{2k - 1}{k + 1}, \frac{k + 3}{k + 1} \right) \]

### Step 3: Substitute into the Plane Equation
Since point \( C \) lies on the plane \( x + y + z = 5 \), we substitute the coordinates of \( C \) into the plane equation:
\[ \frac{-k + 2}{k + 1} + \frac{2k - 1}{k + 1} + \frac{k + 3}{k + 1} = 5 \]

### Step 4: Simplify the Equation
Combining the left-hand side:
\[ \frac{-k + 2 + 2k - 1 + k + 3}{k + 1} = 5 \]
This simplifies to:
\[ \frac{2k + 4}{k + 1} = 5 \]

### Step 5: Cross Multiply
Cross-multiplying gives:
\[ 2k + 4 = 5(k + 1) \]
Expanding the right-hand side:
\[ 2k + 4 = 5k + 5 \]

### Step 6: Rearranging the Equation
Rearranging the equation:
\[ 2k - 5k = 5 - 4 \]
This simplifies to:
\[ -3k = 1 \]
Thus,
\[ k = -\frac{1}{3} \]

### Step 7: Interpret the Result
Since \( k = -\frac{1}{3} \), the ratio in which the line segment is divided is \( 1:3 \) externally.

### Final Answer
The line segment joining the points \( (2, -1, 3) \) and \( (-1, 2, 1) \) is divided by the plane \( x + y + z = 5 \) in the ratio \( 1:3 \) externally.

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