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

Options

A. 24
B. `7/(140)`
C. 47
D. `(40)/7`

Answer

To determine the value of \(4k\) given that the points \((2, 3)\), \((k, 1)\), and \((0, 4)\) are collinear, we can use the concept that the area of the triangle formed by these three points should be zero for them to be collinear.

### Step-by-step Solution:

1. **Identify the points**: We have three points:
- \(A(2, 3)\)
- \(B(k, 1)\)
- \(C(0, 4)\)

2. **Use 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 formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For our points, substituting the coordinates:
\[ A = \frac{1}{2} \left| 2(1 - 4) + k(4 - 3) + 0(3 - 1) \right| \]

3. **Simplify the expression**:
\[ A = \frac{1}{2} \left| 2(-3) + k(1) + 0 \right| \]
\[ A = \frac{1}{2} \left| -6 + k \right| \]

4. **Set the area to zero for collinearity**:
Since the points are collinear, the area must be zero:
\[ \frac{1}{2} \left| -6 + k \right| = 0 \]
This implies:
\[ \left| -6 + k \right| = 0 \]

5. **Solve for \(k\)**:
\[ -6 + k = 0 \implies k = 6 \]

6. **Calculate \(4k\)**:
Now, we need to find \(4k\):
\[ 4k = 4 \times 6 = 24 \]

### Final Answer:
The value of \(4k\) is \(24\).

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