The ratio in which the plane `2x + 3y + 5z = 1`
divides the line segment joining the points (1, 0, 0)
and (1, 3, -5) is

Options

A. `7 : 8`
B. `13 : 12`
C. `15 : 1`
D. `1 : 15`

Answer

To find the ratio in which the plane \(2x + 3y + 5z = 1\) divides the line segment joining the points \(A(1, 0, 0)\) and \(B(1, 3, -5)\), we can follow these steps:

### Step 1: Identify the coordinates of points A and B
The coordinates of the points are:
- \(A(1, 0, 0)\)
- \(B(1, 3, -5)\)

### Step 2: Use the section formula
Let the point \(R\) divide the line segment \(AB\) in the ratio \(\lambda : 1\). According to the section formula, the coordinates of point \(R\) can be expressed as:
\[ R = \left( \frac{\lambda x_2 + 1 \cdot x_1}{\lambda + 1}, \frac{\lambda y_2 + 1 \cdot y_1}{\lambda + 1}, \frac{\lambda z_2 + 1 \cdot z_1}{\lambda + 1} \right) \]
Substituting the coordinates of \(A\) and \(B\):
\[ R = \left( \frac{\lambda \cdot 1 + 1 \cdot 1}{\lambda + 1}, \frac{\lambda \cdot 3 + 1 \cdot 0}{\lambda + 1}, \frac{\lambda \cdot (-5) + 1 \cdot 0}{\lambda + 1} \right) \]
This simplifies to:
\[ R = \left( \frac{1}{\lambda + 1}, \frac{3\lambda}{\lambda + 1}, \frac{-5\lambda}{\lambda + 1} \right) \]

### Step 3: Substitute R into the plane equation
Since point \(R\) lies on the plane \(2x + 3y + 5z = 1\), we substitute the coordinates of \(R\) into the plane equation:
\[ 2\left(\frac{1}{\lambda + 1}\right) + 3\left(\frac{3\lambda}{\lambda + 1}\right) + 5\left(\frac{-5\lambda}{\lambda + 1}\right) = 1 \]
This simplifies to:
\[ \frac{2 + 9\lambda - 25\lambda}{\lambda + 1} = 1 \]
\[ \frac{2 - 16\lambda}{\lambda + 1} = 1 \]

### Step 4: Cross-multiply and solve for \(\lambda\)
Cross-multiplying gives:
\[ 2 - 16\lambda = \lambda + 1 \]
Rearranging this gives:
\[ 2 - 1 = 16\lambda + \lambda \]
\[ 1 = 17\lambda \]
Thus,
\[ \lambda = \frac{1}{17} \]

### Step 5: Find the ratio
The ratio in which the plane divides the line segment \(AB\) is \(\lambda : 1 = \frac{1}{17} : 1\), which can be expressed as:
\[ 1 : 17 \]

### Conclusion
Thus, the ratio in which the plane divides the line segment joining the points \(A(1, 0, 0)\) and \(B(1, 3, -5)\) is \(1 : 17\).