If the points` (2,-3),(k,-1)` and `(0,4)` are collinear, then find the value of` 4 k`.

Options

A. 4
B. `7/140`
C. 47
D. `40/7`

Answer

To determine the value of \( 4k \) when the points \( (2, -3) \), \( (k, -1) \), and \( (0, 4) \) are collinear, we can use the area of the triangle formed by these points. The area will be zero if the points are collinear.

### 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 \( (x_3, y_3) \) 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 \( (2, -3) \), \( (k, -1) \), and \( (0, 4) \), we have:
- \( (x_1, y_1) = (2, -3) \)
- \( (x_2, y_2) = (k, -1) \)
- \( (x_3, y_3) = (0, 4) \)

2. **Substitute the Points into the Area Formula**:
\[ A = \frac{1}{2} \left| 2(-1 - 4) + k(4 + 3) + 0(-3 + 1) \right| \]
Simplifying this, we get:
\[ A = \frac{1}{2} \left| 2(-5) + k(7) + 0 \right| \]
\[ A = \frac{1}{2} \left| -10 + 7k \right| \]

3. **Set the Area to Zero**:
Since the points are collinear, the area must be zero:
\[ \frac{1}{2} \left| -10 + 7k \right| = 0 \]
This implies:
\[ \left| -10 + 7k \right| = 0 \]
Therefore:
\[ -10 + 7k = 0 \]

4. **Solve for \( k \)**:
Rearranging gives:
\[ 7k = 10 \]
\[ k = \frac{10}{7} \]

5. **Calculate \( 4k \)**:
Now, we find \( 4k \):
\[ 4k = 4 \times \frac{10}{7} = \frac{40}{7} \]

### Final Answer:
Thus, the value of \( 4k \) is:
\[ \boxed{\frac{40}{7}} \]