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

Options

A. a. `3:5`
B. b. `4:3`
C. c. `2:3`
D. d. none of these

Answer

To determine the ratio in which the line \(y - x + 2 = 0\) divides the line segment joining the points \((3, -1)\) and \((8, 9)\), we can use the section formula.

### Step-by-Step Solution:

1. **Identify the Points**:
Let \(A(3, -1)\) and \(B(8, 9)\) be the two points.

2. **Equation of the Line**:
The equation of the line is given as \(y - x + 2 = 0\). We can rearrange this to the form \(y = x - 2\).

3. **Assume the Ratio**:
Let the line divide the segment \(AB\) in the ratio \(k:1\). According to the section formula, the coordinates of the point \(C\) that divides the line segment \(AB\) in the ratio \(k:1\) are given by:
\[ C\left(\frac{k \cdot x_2 + 1 \cdot x_1}{k + 1}, \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1}\right) \]
where \(A(x_1, y_1) = (3, -1)\) and \(B(x_2, y_2) = (8, 9)\).

4. **Substituting the Coordinates**:
Substituting the coordinates of points \(A\) and \(B\) into the section formula:
\[ C\left(\frac{k \cdot 8 + 1 \cdot 3}{k + 1}, \frac{k \cdot 9 + 1 \cdot (-1)}{k + 1}\right) = C\left(\frac{8k + 3}{k + 1}, \frac{9k - 1}{k + 1}\right) \]

5. **Substituting into the Line Equation**:
Since point \(C\) lies on the line \(y - x + 2 = 0\), we can substitute the coordinates of \(C\) into this equation:
\[ \frac{9k - 1}{k + 1} - \frac{8k + 3}{k + 1} + 2 = 0 \]

6. **Simplifying the Equation**:
Combine the fractions:
\[ \frac{(9k - 1) - (8k + 3) + 2(k + 1)}{k + 1} = 0 \]
This simplifies to:
\[ \frac{9k - 1 - 8k - 3 + 2k + 2}{k + 1} = 0 \]
which simplifies to:
\[ \frac{3k - 2}{k + 1} = 0 \]
Therefore, \(3k - 2 = 0\) leads to \(k = \frac{2}{3}\).

7. **Finding the Ratio**:
The ratio in which the line divides the segment is \(k:1 = \frac{2}{3}:1\). This can be expressed as \(2:3\).

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