The ratio in which the point (5, 4) divides the line joining points (2, 1) and (7,6 ) is…..

Options

A. `3 : 2`
B. `2 : 3`
C. `1 : 3`
D. `2 : 3`

Answer

To find the ratio in which the point (5, 4) divides the line joining the points (2, 1) and (7, 6), we will use the section formula.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( A(2, 1) \) and \( B(7, 6) \) be the two points.
- Let \( C(5, 4) \) be the point that divides the line segment \( AB \).

2. **Assume the Ratio**:
- Assume the point \( C \) divides the line segment \( AB \) in the ratio \( m:n \). We can express this as \( m:n = \lambda:1 \) where \( \lambda \) is the unknown we need to find.

3. **Use the Section Formula**:
- The section formula states that if a point \( C(x, y) \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

4. **Set Up the Equations**:
- For the x-coordinates:
\[ 5 = \frac{7\lambda + 2 \cdot 1}{\lambda + 1} \]
- For the y-coordinates:
\[ 4 = \frac{6\lambda + 1 \cdot 1}{\lambda + 1} \]

5. **Solve for \( \lambda \)**:
- We will solve the first equation:
\[ 5(\lambda + 1) = 7\lambda + 2 \]
\[ 5\lambda + 5 = 7\lambda + 2 \]
\[ 5 - 2 = 7\lambda - 5\lambda \]
\[ 3 = 2\lambda \]
\[ \lambda = \frac{3}{2} \]

6. **Express the Ratio**:
- The ratio \( m:n = \lambda:1 = \frac{3}{2}:1 \).
- To express this in standard form, multiply both parts by 2:
\[ 3:2 \]

7. **Final Answer**:
- The point (5, 4) divides the line segment joining (2, 1) and (7, 6) in the ratio \( 3:2 \).