In what ratio , the line joining (-1, 1) and ( 5, 7) is divide by the line 2 x + y = 4?

Options

A. ` 5 : 13`
B. ` 5 : 2 `
C. ` 1 : 3 `
D. ` 4 : 7`

Answer

To find the ratio in which the line joining the points (-1, 1) and (5, 7) is divided by the line \(2x + y = 4\), we can follow these steps:

### Step 1: Identify the points and the line equation
The points are \(A(-1, 1)\) and \(B(5, 7)\). The line equation is given as \(2x + y = 4\).

### Step 2: Assume the ratio
Let the line segment \(AB\) be divided by the line \(2x + y = 4\) in the ratio \(k:1\). We can express the coordinates of the point \(P\) that divides \(AB\) in this ratio using the section formula:
\[ P\left(\frac{k \cdot x_2 + x_1}{k + 1}, \frac{k \cdot y_2 + y_1}{k + 1}\right) \]
where \((x_1, y_1) = (-1, 1)\) and \((x_2, y_2) = (5, 7)\).

### Step 3: Substitute the coordinates
Substituting the coordinates into the formula gives:
\[ P\left(\frac{k \cdot 5 + (-1)}{k + 1}, \frac{k \cdot 7 + 1}{k + 1}\right) \]
This simplifies to:
\[ P\left(\frac{5k - 1}{k + 1}, \frac{7k + 1}{k + 1}\right) \]

### Step 4: Substitute into the line equation
Since point \(P\) lies on the line \(2x + y = 4\), we substitute the coordinates of \(P\) into the line equation:
\[ 2\left(\frac{5k - 1}{k + 1}\right) + \left(\frac{7k + 1}{k + 1}\right) = 4 \]

### Step 5: Clear the fractions
Multiply through by \(k + 1\) to eliminate the denominator:
\[ 2(5k - 1) + (7k + 1) = 4(k + 1) \]
Expanding gives:
\[ 10k - 2 + 7k + 1 = 4k + 4 \]

### Step 6: Combine like terms
Combine all terms:
\[ 17k - 1 = 4k + 4 \]
Rearranging gives:
\[ 17k - 4k = 4 + 1 \]
\[ 13k = 5 \]

### Step 7: Solve for \(k\)
Dividing both sides by 13 yields:
\[ k = \frac{5}{13} \]

### Step 8: Determine the ratio
The ratio in which the line divides the segment \(AB\) is \(k:1\), which is:
\[ \frac{5}{13}:1 = 5:13 \]

Thus, the line divides the segment joining the points (-1, 1) and (5, 7) in the ratio **5:13**.

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