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 can use the section formula.
### Step-by-Step Solution:
1. **Identify the Coordinates and Ratio**:
- Let \( A(2, 6) \) and \( B(5, 1) \).
- The point \( C(k, 4) \) divides \( AB \) in the ratio \( 2:3 \).
2. **Apply the Section Formula**:
The section formula states that if a point \( C(x, y) \) divides the line segment joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then:
\[
x = \frac{m \cdot x_2 + n \cdot x_1}{m + n}
\]
\[
y = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
\]
3. **Substituting the Values**:
Here, \( m = 2 \), \( n = 3 \), \( x_1 = 2 \), \( y_1 = 6 \), \( x_2 = 5 \), and \( y_2 = 1 \).
- For the x-coordinate:
\[
k = \frac{2 \cdot 5 + 3 \cdot 2}{2 + 3}
\]
4. **Calculate the x-coordinate**:
\[
k = \frac{10 + 6}{5} = \frac{16}{5}
\]
5. **Verify the y-coordinate**:
The y-coordinate of point \( C \) is given as \( 4 \), so we can also verify using the y-coordinate:
\[
4 = \frac{2 \cdot 1 + 3 \cdot 6}{2 + 3}
\]
\[
4 = \frac{2 + 18}{5} = \frac{20}{5} = 4
\]
This confirms that the calculations are consistent.
6. **Final Answer**:
Thus, the value of \( k \) is:
\[
k = \frac{16}{5}
\]
