If the point C(k, 4) divides the join of points A(2, 6) and B(5, 1) in the ratio 2:3, then the value of k is:

Options

A. 16
B. `28/5`
C. `16/5`
D. `8/5`

Answer

To find the value of \( k \) for the point \( C(k, 4) \) that divides the line segment joining points \( A(2, 6) \) and \( B(5, 1) \) in the ratio \( 2:3 \), we will use the section formula.

### Step-by-Step Solution:

1. **Identify the Coordinates and Ratio**:
- Let \( A = (x_1, y_1) = (2, 6) \)
- Let \( B = (x_2, y_2) = (5, 1) \)
- The ratio in which point \( C \) divides \( AB \) is \( m:n = 2:3 \).

2. **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 the coordinates of point \( C \) are given by:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

3. **Substituting Values for \( x \)**:
- Here, \( m = 2 \), \( n = 3 \), \( x_1 = 2 \), \( x_2 = 5 \).
- Substitute into the formula for \( x \):
\[ k = \frac{2 \cdot 5 + 3 \cdot 2}{2 + 3} \]
\[ k = \frac{10 + 6}{5} \]
\[ k = \frac{16}{5} \]

4. **Substituting Values for \( y \)**:
- Now, we can also check the \( y \)-coordinate:
- Using \( y_1 = 6 \) and \( y_2 = 1 \):
\[ y = \frac{2 \cdot 1 + 3 \cdot 6}{2 + 3} \]
\[ y = \frac{2 + 18}{5} \]
\[ y = \frac{20}{5} = 4 \]
- This confirms that the \( y \)-coordinate is indeed \( 4 \).

5. **Conclusion**:
- Thus, the value of \( k \) is \( \frac{16}{5} \).

### Final Answer:
The value of \( k \) is \( \frac{16}{5} \).

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