Step 1: Understanding the Concept:
When a line \( ax + by + c = 0 \) divides the line segment joining two points \( (x_{1}, y_{1}) \) and \( (x_{2}, y_{2}) \), the ratio \( m:n \) in which it divides the segment can be found using the direct ratio formula.
: Key Formula or Approach:
The ratio \( k \) is given by:
\[ k = -\frac{ax_{1} + by_{1} + c}{ax_{2} + by_{2} + c} \]
If \( k \) is positive, the division is internal. If \( k \) is negative, the division is external.
Step 2: Detailed Explanation:
Given points: \( (x_{1}, y_{1}) = (2, 1) \) and \( (x_{2}, y_{2}) = (0, -2) \).
Given line: \( 2x - 3y + 4 = 0 \).
Using the ratio formula:
\[ \text{Ratio} = -\left[ \frac{2(2) - 3(1) + 4}{2(0) - 3(-2) + 4} \right] \]
Simplify the numerator:
\[ \text{Numerator} = 4 - 3 + 4 = 5 \]
Simplify the denominator:
\[ \text{Denominator} = 0 + 6 + 4 = 10 \]
Calculate the final ratio:
\[ \text{Ratio} = -\frac{5}{10} = -\frac{1}{2} \]
The ratio is \(-1:2\), which indicates external division.
Step 3: Final Answer:
The line divides the segment in the ratio \(-1:2\).