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

Explanation

To find the ratio in which the plane \( x + y + z = \frac{1}{5} \) divides the line joining the points \( A(3, 1, 4) \) and \( B(4, 2, 5) \), we can follow these steps:

### Step 1: Assign coordinates and define the ratio
Let the line segment \( AB \) be divided by the plane at point \( C \) in the ratio \( k:1 \). The coordinates of point \( C \) can be expressed using the section formula:
\[ C\left( \frac{k \cdot x_2 + 1 \cdot x_1}{k + 1}, \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1}, \frac{k \cdot z_2 + 1 \cdot z_1}{k + 1} \right) \]
where \( A(3, 1, 4) \) is \( (x_1, y_1, z_1) \) and \( B(4, 2, 5) \) is \( (x_2, y_2, z_2) \).

### Step 2: Calculate the coordinates of point C
Using the coordinates of points \( A \) and \( B \):
- \( x_1 = 3, y_1 = 1, z_1 = 4 \)
- \( x_2 = 4, y_2 = 2, z_2 = 5 \)

The coordinates of point \( C \) are:
\[ C\left( \frac{4k + 3}{k + 1}, \frac{2k + 1}{k + 1}, \frac{5k + 4}{k + 1} \right) \]

### Step 3: Substitute into the plane equation
Since point \( C \) lies on the plane \( x + y + z = \frac{1}{5} \), we substitute the coordinates of \( C \) into the plane equation:
\[ \frac{4k + 3}{k + 1} + \frac{2k + 1}{k + 1} + \frac{5k + 4}{k + 1} = \frac{1}{5} \]

### Step 4: Combine the fractions
Combining the left-hand side:
\[ \frac{(4k + 3) + (2k + 1) + (5k + 4)}{k + 1} = \frac{11k + 8}{k + 1} \]
Thus, we have:
\[ \frac{11k + 8}{k + 1} = \frac{1}{5} \]

### Step 5: Cross-multiply to solve for k
Cross-multiplying gives:
\[ 5(11k + 8) = 1(k + 1) \]
Expanding both sides:
\[ 55k + 40 = k + 1 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 55k - k = 1 - 40 \]
\[ 54k = -39 \]
\[ k = -\frac{39}{54} = -\frac{13}{18} \]

### Step 7: Determine the ratio
The ratio in which the plane divides the line is \( k:1 \), which simplifies to:
\[ \text{Ratio} = -\frac{13}{18} : 1 = 13 : 18 \]

### Final Answer
The ratio in which the plane divides the line is \( 13:18 \).

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