Find the value of k if the point `(2,3) ,B(4,k) and C(6,-3)` are collinear.

Answer

To find the value of \( k \) such that the points \( A(2, 3) \), \( B(4, k) \), and \( C(6, -3) \) are collinear, we can use the concept that the area of the triangle formed by these three points should be zero.

### Step-by-Step Solution:

1. **Set Up the Area Formula**:
The area \( A \) of a triangle formed by three points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the determinant:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For points \( A(2, 3) \), \( B(4, k) \), and \( C(6, -3) \), we substitute:
\[ A = \frac{1}{2} \left| 2(k - (-3)) + 4(-3 - 3) + 6(3 - k) \right| \]

2. **Simplify the Expression**:
Substitute the coordinates into the area formula:
\[ A = \frac{1}{2} \left| 2(k + 3) + 4(-6) + 6(3 - k) \right| \]
Simplifying further:
\[ A = \frac{1}{2} \left| 2k + 6 - 24 + 18 - 6k \right| \]
\[ A = \frac{1}{2} \left| -4k + 0 \right| \]
\[ A = \frac{1}{2} \left| -4k \right| = 2|k| \]

3. **Set the Area to Zero**:
For the points to be collinear, the area must be zero:
\[ 2|k| = 0 \]
This implies:
\[ |k| = 0 \]

4. **Solve for \( k \)**:
Since the absolute value of \( k \) is zero, we conclude:
\[ k = 0 \]

### Final Answer:
The value of \( k \) for which the points \( A(2, 3) \), \( B(4, k) \), and \( C(6, -3) \) are collinear is:
\[ \boxed{0} \]