The points A (3, -2) , B (1,4) and C (-2, x) are collinear. What is the value of x ?

Options

A. 13
B. `-2`
C. 5
D. 3

Answer

To find the value of \( x \) such that the points \( A(3, -2) \), \( B(1, 4) \), and \( C(-2, x) \) 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 coordinates of the points:**
- Let \( A(x_1, y_1) = (3, -2) \)
- Let \( B(x_2, y_2) = (1, 4) \)
- Let \( C(x_3, y_3) = (-2, x) \)

2. **Use the area formula for collinear points:**
The area \( A \) of the triangle formed by points \( A \), \( B \), and \( C \) 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 the points to be collinear, this area must equal zero:
\[ x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) = 0 \]

3. **Substitute the coordinates into the formula:**
\[ 3(4 - x) + 1(x - (-2)) + (-2)(-2 - 4) = 0 \]
Simplifying this gives:
\[ 3(4 - x) + 1(x + 2) - 2(-6) = 0 \]

4. **Distributing and simplifying:**
\[ 12 - 3x + x + 2 + 12 = 0 \]
Combine like terms:
\[ 12 + 2 + 12 - 3x + x = 0 \]
\[ 26 - 2x = 0 \]

5. **Solve for \( x \):**
\[ -2x = -26 \]
\[ x = \frac{-26}{-2} = 13 \]

### Conclusion:
The value of \( x \) is \( 13 \).