The line `x+y=4` divides the line joining the points (-1,1) and (5,7) in the ratio

Options

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

Answer

To solve the problem, we need to find the ratio in which the line \(x + y = 4\) divides the line segment joining the points \((-1, 1)\) and \((5, 7)\).

### Step 1: Identify the points
Let the points be:
- \(P(-1, 1)\)
- \(Q(5, 7)\)

### Step 2: Use the section formula
Let \(R(x, y)\) be the point that divides the line segment \(PQ\) in the ratio \(m:n\). According to the section formula, the coordinates of point \(R\) can be expressed as:
\[ R\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
where \(P(x_1, y_1) = (-1, 1)\) and \(Q(x_2, y_2) = (5, 7)\).

### Step 3: Substitute the coordinates
Let’s assume the ratio is \(m:1\) (i.e., \(m\) is the unknown ratio we want to find). Then the coordinates of point \(R\) become:
\[ R\left(\frac{m \cdot 5 + 1 \cdot (-1)}{m + 1}, \frac{m \cdot 7 + 1 \cdot 1}{m + 1}\right) = R\left(\frac{5m - 1}{m + 1}, \frac{7m + 1}{m + 1}\right) \]

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

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

### Step 6: Cross-multiply and solve for \(m\)
Cross-multiplying gives:
\[ 12m = 4(m + 1) \]
Expanding the right side:
\[ 12m = 4m + 4 \]
Rearranging gives:
\[ 12m - 4m = 4 \implies 8m = 4 \implies m = \frac{1}{2} \]

### Step 7: Determine the ratio
The ratio in which the line divides the segment \(PQ\) is \(m:1 = \frac{1}{2}:1\), which simplifies to:
\[ 1:2 \]

### Final Answer
Thus, 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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