In what ratio does the line x - y - 2 = 0 divide the line segment joining the points (3.-1) and (8,9) ? Also, find the co-ordinates of the point of intersection.

Answer

To solve the problem, we need to determine the ratio in which the line \( x - y - 2 = 0 \) divides the line segment joining the points \( (3, -1) \) and \( (8, 9) \), and also find the coordinates of the point of intersection.

### Step 1: Identify the coordinates of the points
We have two points:
- Point A: \( (3, -1) \)
- Point B: \( (8, 9) \)

### Step 2: Use the section formula
Let the point of intersection be \( P(a, b) \). According to the section formula, if a line divides the segment joining points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \), then the coordinates of the point of intersection \( P \) can be given by:
\[ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
Substituting the coordinates of points A and B:
- \( x_1 = 3 \), \( y_1 = -1 \)
- \( x_2 = 8 \), \( y_2 = 9 \)

Thus, the coordinates of point \( P \) become:
\[ P\left( \frac{8m + 3n}{m+n}, \frac{9m - n}{m+n} \right) \]

### Step 3: Substitute into the line equation
Since point \( P \) lies on the line \( x - y - 2 = 0 \), we can substitute the coordinates of \( P \) into this equation:
\[ \frac{8m + 3n}{m+n} - \frac{9m - n}{m+n} - 2 = 0 \]
This simplifies to:
\[ \frac{(8m + 3n) - (9m - n)}{m+n} - 2 = 0 \]
\[ \frac{8m + 3n - 9m + n}{m+n} - 2 = 0 \]
\[ \frac{-m + 4n}{m+n} - 2 = 0 \]

### Step 4: Clear the fraction
Multiply through by \( m+n \):
\[ -m + 4n - 2(m+n) = 0 \]
\[ -m + 4n - 2m - 2n = 0 \]
\[ -3m + 2n = 0 \]
This gives us:
\[ 3m = 2n \quad \Rightarrow \quad \frac{m}{n} = \frac{2}{3} \]

### Step 5: Find the coordinates of the point of intersection
Let \( m = 2 \) and \( n = 3 \). Now substitute these values back into the section formula:
\[ P\left( \frac{8(2) + 3(3)}{2+3}, \frac{9(2) - 3}{2+3} \right) \]
Calculating the x-coordinate:
\[ P_x = \frac{16 + 9}{5} = \frac{25}{5} = 5 \]
Calculating the y-coordinate:
\[ P_y = \frac{18 - 3}{5} = \frac{15}{5} = 3 \]
Thus, the coordinates of the point of intersection \( P \) are \( (5, 3) \).

### Final Answer
The ratio in which the line divides the segment is \( 2:3 \) and the coordinates of the point of intersection are \( (5, 3) \).