In what ratio does the line x - y - 2= 0 divide the line segment joining the points A(3, -1) and B(8, 9)?

Explanation

To find the ratio in which the line \( x - y - 2 = 0 \) divides the line segment joining the points \( A(3, -1) \) and \( B(8, 9) \), we can follow these steps:

### Step 1: Understand the problem
We need to find the point \( P \) that divides the line segment \( AB \) in the ratio \( \lambda:1 \) and lies on the line \( x - y - 2 = 0 \).

### Step 2: Use the section formula
The coordinates of point \( P \) that divides the segment \( AB \) in the ratio \( \lambda:1 \) can be calculated using the section formula:
\[ P\left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right) \]
where \( A(x_1, y_1) = (3, -1) \) and \( B(x_2, y_2) = (8, 9) \).

### Step 3: Substitute values into the section formula
Substituting \( m = \lambda \), \( n = 1 \), \( x_1 = 3 \), \( y_1 = -1 \), \( x_2 = 8 \), and \( y_2 = 9 \):
\[ P\left( \frac{8\lambda + 3}{\lambda + 1}, \frac{9\lambda - 1}{\lambda + 1} \right) \]

### Step 4: Set the coordinates of point \( P \) on the line
Since point \( P \) lies on the line \( x - y - 2 = 0 \), we can substitute the coordinates of \( P \) into the line equation:
\[ \frac{8\lambda + 3}{\lambda + 1} - \frac{9\lambda - 1}{\lambda + 1} - 2 = 0 \]

### Step 5: Simplify the equation
Combining the fractions:
\[ \frac{(8\lambda + 3) - (9\lambda - 1) - 2(\lambda + 1)}{\lambda + 1} = 0 \]
This simplifies to:
\[ 8\lambda + 3 - 9\lambda + 1 - 2\lambda - 2 = 0 \]
\[ -3\lambda + 2 = 0 \]

### Step 6: Solve for \( \lambda \)
Rearranging gives:
\[ 3\lambda = 2 \implies \lambda = \frac{2}{3} \]

### Step 7: Find the ratio
The ratio in which the line divides the segment \( AB \) is \( \lambda:1 \), which is:
\[ \frac{2}{3}:1 \implies 2:3 \]

### Final Answer
The line \( x - y - 2 = 0 \) divides the line segment joining the points \( A(3, -1) \) and \( B(8, 9) \) in the ratio \( 2:3 \).

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