Find the ratio in which the line joining the points (2, 3, 5) and (3, 4, 1) is divided by the plane
`x - 2y + z =5.`

Answer

To find the ratio in which the line joining the points \( A(2, 3, 5) \) and \( B(3, 4, 1) \) is divided by the plane given by the equation \( x - 2y + z = 5 \), we will follow these steps:

### Step 1: Find the direction ratios of the line segment AB
The direction ratios of the line segment joining points \( A \) and \( B \) can be calculated as follows:

\[ \text{Direction ratios} = (3 - 2, 4 - 3, 1 - 5) = (1, 1, -4) \]

### Step 2: Write the parametric equations of the line
Using the point \( A(2, 3, 5) \) as the starting point, we can express the parametric equations of the line as:

\[ x = 2 + t, \quad y = 3 + t, \quad z = 5 - 4t \]

where \( t \) is a parameter.

### Step 3: Substitute the parametric equations into the plane equation
We need to find the point of intersection of the line with the plane \( x - 2y + z = 5 \). Substituting the parametric equations into the plane equation:

\[ (2 + t) - 2(3 + t) + (5 - 4t) = 5 \]

### Step 4: Simplify the equation
Now, simplifying the equation:

\[ 2 + t - 6 - 2t + 5 - 4t = 5 \]

Combining like terms gives:

\[ 1 - 5t = 5 \]

### Step 5: Solve for \( t \)
Rearranging the equation:

\[ -5t = 5 - 1 \implies -5t = 4 \implies t = -\frac{4}{5} \]

### Step 6: Find the coordinates of the intersection point
Now substituting \( t = -\frac{4}{5} \) back into the parametric equations to find the coordinates of the intersection point \( R \):

\[ x = 2 - \frac{4}{5} = \frac{10}{5} - \frac{4}{5} = \frac{6}{5} \]

\[ y = 3 - \frac{4}{5} = \frac{15}{5} - \frac{4}{5} = \frac{11}{5} \]

\[ z = 5 + \frac{16}{5} = \frac{25}{5} + \frac{16}{5} = \frac{41}{5} \]

Thus, the coordinates of point \( R \) are \( \left( \frac{6}{5}, \frac{11}{5}, \frac{41}{5} \right) \).

### Step 7: Use the section formula to find the ratio
Let the point \( R \) divide the line segment \( AB \) in the ratio \( k:1 \). According to the section formula, we have:

\[ \frac{6}{5} = \frac{3k + 2}{k + 1} \]

Cross-multiplying gives:

\[ 6(k + 1) = 5(3k + 2) \]

Expanding both sides:

\[ 6k + 6 = 15k + 10 \]

Rearranging gives:

\[ 15k - 6k = 10 - 6 \implies 9k = 4 \implies k = \frac{4}{9} \]

### Step 8: Determine the ratio
Since \( k = \frac{4}{9} \), the ratio in which the line is divided is \( 4:9 \). However, since the value of \( k \) is negative, it indicates that the division is external.

### Final Answer
The line joining the points \( (2, 3, 5) \) and \( (3, 4, 1) \) is divided by the plane in the ratio \( 4:9 \) externally.

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