Find the ratio in which the join of points (3, -1) and (8, 9) is divided by the line y-x+2=0.

Answer

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

### Step 1: Write the equation of the line in slope-intercept form.
The given line is \( y - x + 2 = 0 \). Rearranging this, we get:
\[ y = x - 2 \]

### Step 2: Find the coordinates of the points A and B.
The coordinates of point \( A \) are \( (3, -1) \) and the coordinates of point \( B \) are \( (8, 9) \).

### Step 3: Assume the coordinates of the point P that divides AB.
Let the point \( P \) that divides the line segment \( AB \) in the ratio \( k:1 \) have coordinates \( (x, y) \). According to the section formula, the coordinates of point \( P \) can be expressed as:
\[ P\left(\frac{3 \cdot 1 + 8 \cdot k}{k + 1}, \frac{-1 \cdot 1 + 9 \cdot k}{k + 1}\right) \]

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

### Step 5: Simplify the equation.
First, simplify the right-hand side:
\[ \frac{3 + 8k}{k + 1} - 2 = \frac{3 + 8k - 2(k + 1)}{k + 1} = \frac{3 + 8k - 2k - 2}{k + 1} = \frac{6k + 1}{k + 1} \]
Now, we equate the two sides:
\[ \frac{-1 + 9k}{k + 1} = \frac{6k + 1}{k + 1} \]
Since the denominators are the same, we can equate the numerators:
\[ -1 + 9k = 6k + 1 \]

### Step 6: Solve for k.
Rearranging gives:
\[ 9k - 6k = 1 + 1 \implies 3k = 2 \implies k = \frac{2}{3} \]

### Step 7: Write the final ratio.
Since we assumed the ratio was \( k:1 \), we have:
\[ \text{Ratio} = \frac{2}{3}:1 = 2:3 \]

### Conclusion
The ratio in which the line segment joining the points \( (3, -1) \) and \( (8, 9) \) is divided by the line \( y - x + 2 = 0 \) is \( 2:3 \).

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