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 \( P(2, 3, 5) \) and \( Q(3, 4, 1) \) is divided by the plane \( x - 2y + z = 5 \), we can follow these steps:

### Step 1: Determine the coordinates of the points
Let:
- \( P = (2, 3, 5) \)
- \( Q = (3, 4, 1) \)

### Step 2: Use the section formula
Let the point \( R \) divide the line segment \( PQ \) in the ratio \( m:n \). The coordinates of point \( R \) can be expressed using the section formula:
\[ R = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]
where \( (x_1, y_1, z_1) = (2, 3, 5) \) and \( (x_2, y_2, z_2) = (3, 4, 1) \).

### Step 3: Substitute the coordinates into the section formula
Substituting the coordinates of \( P \) and \( Q \):
\[ R = \left( \frac{m \cdot 3 + n \cdot 2}{m+n}, \frac{m \cdot 4 + n \cdot 3}{m+n}, \frac{m \cdot 1 + n \cdot 5}{m+n} \right) \]
This gives us:
\[ R = \left( \frac{3m + 2n}{m+n}, \frac{4m + 3n}{m+n}, \frac{m + 5n}{m+n} \right) \]

### Step 4: Substitute \( R \) into the plane equation
Since point \( R \) lies on the plane \( x - 2y + z = 5 \), we substitute the coordinates of \( R \) into the plane equation:
\[ \frac{3m + 2n}{m+n} - 2 \cdot \frac{4m + 3n}{m+n} + \frac{m + 5n}{m+n} = 5 \]
Multiplying through by \( m+n \) to eliminate the denominator:
\[ 3m + 2n - 2(4m + 3n) + (m + 5n) = 5(m+n) \]

### Step 5: Simplify the equation
Expanding and simplifying:
\[ 3m + 2n - 8m - 6n + m + 5n = 5m + 5n \]
Combining like terms:
\[ (3m - 8m + m) + (2n - 6n + 5n) = 5m + 5n \]
This simplifies to:
\[ -4m + n = 5m + 5n \]
Rearranging gives:
\[ -4m - 5m = 5n - n \]
\[ -9m = 4n \]
Thus, we have:
\[ \frac{m}{n} = -\frac{4}{9} \]

### Step 6: Determine the ratio
The ratio \( m:n \) is \( -4:9 \). Since the ratio is negative, it indicates that the division is external. Therefore, we can express the ratio as:
\[ 4:9 \]

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

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