Question

Determine the ratio in which the line 3x + y - 9 = 0 divides the line segment joining the points (1, 3) and (2, 5). Find the point of intersection.

Hint:

To quickly find the dividing ratio \( k \) in which a line \( Ax + By + C = 0 \) divides the segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \), use the formula:
\[ k = -\frac{Ax_1 + By_1 + C}{Ax_2 + By_2 + C} \]
Substitute the values:
\[ k = -\frac{3(1) + 3 - 9}{3(2) + 5 - 9} = -\frac{-3}{2} = \frac{3}{2} \]
This formula gives the ratio \( 3 : 2 \) instantly and minimizes calculation errors.

Answer 1

Step 1: Understanding the Question:
The topic of this question is Coordinate Geometry (Section Formula and Line division).
We are given two points \( A(1, 3) \) and \( B(2, 5) \) representing a line segment.
A third line, \( 3x + y - 9 = 0 \), intersects this line segment.
We need to find the ratio in which the line divides the segment \( AB \) and the exact coordinates of the intersection point.

Step 2: Key Formula or Approach:
- Let the line divide the segment \( AB \) in the ratio \( k : 1 \) at some point \( R \).
- Use the section formula to find the coordinates of \( R(x, y) \):
\[ x = \frac{kx_2 + x_1}{k + 1}, \quad y = \frac{ky_2 + y_1}{k + 1} \]
- Since \( R \) lies on the dividing line, substitute these coordinate expressions into the equation of the line \( 3x + y - 9 = 0 \) and solve for \( k \).

Step 3: Detailed Explanation:
1. Let the dividing point on the segment \( AB \) be \( R \), and let the dividing ratio be \( k : 1 \).
Given points: \( A(x_1, y_1) = (1, 3) \) and \( B(x_2, y_2) = (2, 5) \).
2. Use the section formula to write the coordinates of point \( R \):
\[ x = \frac{k(2) + 1(1)}{k + 1} = \frac{2k + 1}{k + 1} \]
\[ y = \frac{k(5) + 1(3)}{k + 1} = \frac{5k + 3}{k + 1} \]
So, \( R = \left(\frac{2k + 1}{k + 1}, \frac{5k + 3}{k + 1}\right) \).
3. Since the point \( R \) lies on the line \( 3x + y - 9 = 0 \), its coordinates must satisfy the equation:
\[ 3\left(\frac{2k + 1}{k + 1}\right) + \left(\frac{5k + 3}{k + 1}\right) - 9 = 0 \]
4. Multiply the entire equation by \( k + 1 \) to clear the denominators:
\[ 3(2k + 1) + (5k + 3) - 9(k + 1) = 0 \]
5. Expand the terms:
\[ 6k + 3 + 5k + 3 - 9k - 9 = 0 \]
Combine the \( k \)-terms and constants:
\[ (6k + 5k - 9k) + (3 + 3 - 9) = 0 \]
\[ 2k - 3 = 0 \]
\[ 2k = 3 \implies k = \frac{3}{2} \]
Thus, the dividing ratio is \( \frac{3}{2} : 1 \), which is \( 3 : 2 \).
6. Now, calculate the coordinates of the intersection point \( R \) by substituting \( k = \frac{3}{2} \):
\[ x = \frac{2\left(\frac{3}{2}\right) + 1}{\frac{3}{2} + 1} = \frac{3 + 1}{\frac{5}{2}} = \frac{4}{\frac{5}{2}} = \frac{8}{5} = 1.6 \]
\[ y = \frac{5\left(\frac{3}{2}\right) + 3}{\frac{3}{2} + 1} = \frac{\frac{15}{2} + 3}{\frac{5}{2}} = \frac{\frac{21}{2}}{\frac{5}{2}} = \frac{21}{5} = 4.2 \]
7. Thus, the point of intersection is \(\left(\frac{8}{5}, \frac{21}{5}\right)\) or \((1.6, 4.2)\).

Step 4: Final Answer:
The line divides the segment in the ratio \(3 : 2\), and the coordinates of the point of intersection are \((1.6, 4.2)\).