Find the value of k if the points (2,3) , (5,k) and (6,7) are collinear .

Answer

To find the value of \( k \) such that the points \( (2, 3) \), \( (5, k) \), and \( (6, 7) \) are collinear, we can use the formula for the area of a triangle formed by three points. If the area is zero, then the points are collinear.

### Step-by-Step Solution:

1. **Use the Area Formula**: The area \( A \) of a triangle formed by the points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For our points, we have:
- \( (x_1, y_1) = (2, 3) \)
- \( (x_2, y_2) = (5, k) \)
- \( (x_3, y_3) = (6, 7) \)

2. **Substitute the Points into the Formula**:
\[ A = \frac{1}{2} \left| 2(k - 7) + 5(7 - 3) + 6(3 - k) \right| \]

3. **Simplify the Expression**:
\[ A = \frac{1}{2} \left| 2k - 14 + 5 \times 4 + 6 \times (3 - k) \right| \]
\[ = \frac{1}{2} \left| 2k - 14 + 20 + 18 - 6k \right| \]
\[ = \frac{1}{2} \left| -4k + 24 \right| \]

4. **Set the Area to Zero** (since the points are collinear):
\[ \frac{1}{2} \left| -4k + 24 \right| = 0 \]
This implies:
\[ \left| -4k + 24 \right| = 0 \]

5. **Solve the Absolute Value Equation**:
\[ -4k + 24 = 0 \]
\[ -4k = -24 \]
\[ k = 6 \]

### Final Answer:
The value of \( k \) is \( 6 \).

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