Find the value of k for which the points `(7,-2),(5,1) and (3,k)` are collinear.

Answer

To find the value of \( k \) for which the points \( (7, -2) \), \( (5, 1) \), and \( (3, k) \) are collinear, we can use the formula for the area of a triangle formed by three points. If the area is zero, the points are collinear.

### Step-by-Step Solution:

1. **Identify the points**:
- Let \( A(7, -2) \), \( B(5, 1) \), and \( C(3, k) \).

2. **Use the area formula**:
The area \( A \) of the 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, substituting \( (x_1, y_1) = (7, -2) \), \( (x_2, y_2) = (5, 1) \), and \( (x_3, y_3) = (3, k) \):
\[ A = \frac{1}{2} \left| 7(1 - k) + 5(k + 2) + 3(-2 - 1) \right| \]

3. **Simplify the expression**:
\[ A = \frac{1}{2} \left| 7(1 - k) + 5(k + 2) - 9 \right| \]
Expanding this:
\[ = \frac{1}{2} \left| 7 - 7k + 5k + 10 - 9 \right| \]
\[ = \frac{1}{2} \left| -2k + 8 \right| \]

4. **Set the area to zero**:
Since the points are collinear, the area must be zero:
\[ \frac{1}{2} \left| -2k + 8 \right| = 0 \]
This implies:
\[ \left| -2k + 8 \right| = 0 \]

5. **Solve for \( k \)**:
\[ -2k + 8 = 0 \]
\[ -2k = -8 \]
\[ k = 4 \]

### Final Answer:
The value of \( k \) for which the points \( (7, -2) \), \( (5, 1) \), and \( (3, k) \) are collinear is \( k = 4 \).