The ratio in which the line segment joining (-1,1) and (5,7) is divided by the line `x+y=4` is

Options

A. `1:2` internally
B. `1:2` externally
C. `2:1` internally
D. `2:1` externally

Answer

To solve the problem of finding the ratio in which the line segment 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:
- \(A(-1, 1)\) (Point 1)
- \(B(5, 7)\) (Point 2)

### Step 2: Use the Section Formula
Let the point \(R(x, y)\) divide the segment \(AB\) in the ratio \(m:n\). According to the section formula, the coordinates of point \(R\) can be expressed as:
\[ R\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Here, \(x_1 = -1\), \(y_1 = 1\), \(x_2 = 5\), \(y_2 = 7\), and we will denote the ratio as \(m:n\) where \(m = \lambda\) and \(n = 1\).

### Step 3: Substitute the Coordinates
Substituting the coordinates into the section formula gives:
\[ R\left(\frac{5\lambda - 1}{\lambda + 1}, \frac{7\lambda + 1}{\lambda + 1}\right) \]

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

### Step 5: Simplify the Equation
Combine 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:
\[ 12\lambda = 4(\lambda + 1) \]

### Step 7: Expand and Rearrange
Expanding the right side:
\[ 12\lambda = 4\lambda + 4 \]
Rearranging gives:
\[ 12\lambda - 4\lambda = 4 \]
\[ 8\lambda = 4 \]

### Step 8: Solve for \(\lambda\)
Dividing both sides by 8:
\[ \lambda = \frac{4}{8} = \frac{1}{2} \]

### Step 9: Determine the Ratio
The ratio \(m:n\) is \(\lambda:1\), which is \(\frac{1}{2}:1\). This can be expressed as:
\[ 1:2 \]
Thus, the line segment is divided in the ratio \(1:2\) internally.

### Final Answer
The ratio in which the line segment joining (-1, 1) and (5, 7) is divided by the line \(x + y = 4\) is \(1:2\) internally.

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