In what ratio, the line joining (-1, 1) and ( 5, 7) is divided by the line x + y = 4 ?

Options

A. `2 : 1 `
B. ` 1 : 3 `
C. ` 1 : 2 `
D. `1 : 4 `

Answer

To solve the problem of finding the ratio in which the line joining the points (-1, 1) and (5, 7) is divided by the line \(x + y = 4\), we can follow these steps:

### Step 1: Identify the Points
Let the points be:
- \( A(-1, 1) \)
- \( B(5, 7) \)

### Step 2: Use the Section Formula
We need to find the coordinates of the point \( O \) where the line \( x + y = 4 \) intersects the line segment \( AB \). We will assume that the line divides \( AB \) in the ratio \( \lambda : 1 \).

Using the section formula, the coordinates of point \( O \) can be expressed as:
\[ O\left(\frac{\lambda \cdot x_2 + 1 \cdot x_1}{\lambda + 1}, \frac{\lambda \cdot y_2 + 1 \cdot y_1}{\lambda + 1}\right) \]
where \( (x_1, y_1) = (-1, 1) \) and \( (x_2, y_2) = (5, 7) \).

### Step 3: Calculate the Coordinates of Point O
Substituting the values into the section formula:
- For the x-coordinate:
\[ x = \frac{\lambda \cdot 5 + 1 \cdot (-1)}{\lambda + 1} = \frac{5\lambda - 1}{\lambda + 1} \]
- For the y-coordinate:
\[ y = \frac{\lambda \cdot 7 + 1 \cdot 1}{\lambda + 1} = \frac{7\lambda + 1}{\lambda + 1} \]

### Step 4: Set Up the Equation
Since point \( O \) lies on the line \( x + y = 4 \), we substitute the coordinates into this equation:
\[ \frac{5\lambda - 1}{\lambda + 1} + \frac{7\lambda + 1}{\lambda + 1} = 4 \]

### Step 5: Simplify the Equation
Combining the fractions:
\[ \frac{(5\lambda - 1) + (7\lambda + 1)}{\lambda + 1} = 4 \]
This simplifies to:
\[ \frac{12\lambda}{\lambda + 1} = 4 \]

### Step 6: Cross Multiply
Cross multiplying gives:
\[ 12\lambda = 4(\lambda + 1) \]
Expanding the right side:
\[ 12\lambda = 4\lambda + 4 \]

### Step 7: Solve for Lambda
Rearranging the equation:
\[ 12\lambda - 4\lambda = 4 \]
\[ 8\lambda = 4 \]
\[ \lambda = \frac{4}{8} = \frac{1}{2} \]

### Step 8: Determine the Ratio
The ratio in which the line divides the segment \( AB \) is \( \lambda : 1 = \frac{1}{2} : 1 \), which simplifies to:
\[ 1 : 2 \]

### Final Answer
The line \( x + y = 4 \) divides the line segment joining the points (-1, 1) and (5, 7) in the ratio \( 1 : 2 \).

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