Determine the ratio in which the line y - x + 2 = 0 divides the line segment joining the points (3, -1) and (8, 9).

Answer

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

### Step 1: Identify the Coordinates
Let the coordinates of point \( A \) be \( (x_1, y_1) = (3, -1) \) and the coordinates of point \( B \) be \( (x_2, y_2) = (8, 9) \).

### Step 2: Use the Section Formula
If a point \( D \) divides the line segment \( AB \) in the ratio \( m:n \), then the coordinates of point \( D \) can be expressed using the section formula:
\[ D\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
Here, we will denote \( m \) as \( \lambda \) and \( n \) as \( 1 \) (i.e., the ratio is \( \lambda:1 \)).

### Step 3: Substitute the Coordinates
Substituting the coordinates of points \( A \) and \( B \) into the section formula:
\[ D\left( \frac{\lambda \cdot 8 + 1 \cdot 3}{\lambda + 1}, \frac{\lambda \cdot 9 + 1 \cdot (-1)}{\lambda + 1} \right) \]
This simplifies to:
\[ D\left( \frac{8\lambda + 3}{\lambda + 1}, \frac{9\lambda - 1}{\lambda + 1} \right) \]

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

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

### Step 6: Solve for \( \lambda \)
Setting the numerator equal to zero gives:
\[ 3\lambda - 2 = 0 \implies 3\lambda = 2 \implies \lambda = \frac{2}{3} \]

### Step 7: Determine the Ratio
The ratio in which the line divides the segment \( AB \) is \( \lambda:1 = \frac{2}{3}:1 \), which can be expressed as:
\[ \text{Ratio} = 2:3 \]

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

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