The ratio in which the joining of `(2,1,5) and (3,4,3) ` is divded by the plane `(x + y -z) = (1)/(2) ` is :

Options

A. `3:5`
B. `5:7`
C. `1:3`
D. `4:5`

Answer

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

### Step 1: Determine the coordinates of point C
Let the point \( C \) divide the line segment \( AB \) in the ratio \( \lambda : 1 \). The coordinates of point \( C \) can be expressed using the section formula:

\[ C\left( \frac{\lambda x_2 + x_1}{\lambda + 1}, \frac{\lambda y_2 + y_1}{\lambda + 1}, \frac{\lambda z_2 + z_1}{\lambda + 1} \right) \]

Where:
- \( A(x_1, y_1, z_1) = (2, 1, 5) \)
- \( B(x_2, y_2, z_2) = (3, 4, 3) \)

Substituting the coordinates of points \( A \) and \( B \):

\[ C\left( \frac{\lambda \cdot 3 + 2}{\lambda + 1}, \frac{\lambda \cdot 4 + 1}{\lambda + 1}, \frac{\lambda \cdot 3 + 5}{\lambda + 1} \right) \]

### Step 2: Substitute the coordinates of C into the plane equation
The plane equation is given by:

\[ x + y - z = \frac{1}{2} \]

Substituting the coordinates of point \( C \):

\[ \frac{\lambda \cdot 3 + 2}{\lambda + 1} + \frac{\lambda \cdot 4 + 1}{\lambda + 1} - \frac{\lambda \cdot 3 + 5}{\lambda + 1} = \frac{1}{2} \]

### Step 3: Simplify the equation
Combining the terms on the left-hand side:

\[ \frac{(\lambda \cdot 3 + 2) + (\lambda \cdot 4 + 1) - (\lambda \cdot 3 + 5)}{\lambda + 1} = \frac{1}{2} \]

This simplifies to:

\[ \frac{3\lambda + 4 - 5 + 2}{\lambda + 1} = \frac{1}{2} \]

\[ \frac{3\lambda + 1}{\lambda + 1} = \frac{1}{2} \]

### Step 4: Cross-multiply to solve for λ
Cross-multiplying gives:

\[ 2(3\lambda + 1) = 1(\lambda + 1) \]

Expanding both sides:

\[ 6\lambda + 2 = \lambda + 1 \]

### Step 5: Rearranging the equation
Rearranging gives:

\[ 6\lambda - \lambda = 1 - 2 \]

\[ 5\lambda = -1 \]

\[ \lambda = -\frac{1}{5} \]

### Step 6: Determine the ratio
The ratio in which point \( C \) divides \( AB \) is \( 1 : \lambda \). Since \( \lambda = -\frac{1}{5} \), we can express the ratio as:

\[ 1 : -\frac{1}{5} = 5 : -1 \]

However, since we are interested in the positive ratio, we can take the absolute values:

\[ 5 : 1 \]

Thus, the final answer is:

\[ \text{The ratio is } 5 : 1 \]