If the points (3,-2) , (x,2) and (8,8) are collinear then the value fo x is :

Options

A. 2
B. 5
C. 4
D. 3

Answer

To find the value of \( x \) such that the points \( (3, -2) \), \( (x, 2) \), and \( (8, 8) \) are collinear, we can use the determinant method. The points are collinear if the determinant of the matrix formed by these points is equal to zero.

### Step-by-step Solution:

1. **Set up the determinant**:
We will set up the determinant using the coordinates of the points:
\[ \begin{vmatrix} 3 & -2 & 1 \\ x & 2 & 1 \\ 8 & 8 & 1 \end{vmatrix} = 0 \]

2. **Calculate the determinant**:
We can calculate the determinant using the formula:
\[ \text{Determinant} = 3 \begin{vmatrix} 2 & 1 \\ 8 & 1 \end{vmatrix} - (-2) \begin{vmatrix} x & 1 \\ 8 & 1 \end{vmatrix} + 1 \begin{vmatrix} x & 2 \\ 8 & 8 \end{vmatrix} \]

Now, we will calculate each of the 2x2 determinants:
- For \( \begin{vmatrix} 2 & 1 \\ 8 & 1 \end{vmatrix} \):
\[ = (2 \cdot 1) - (1 \cdot 8) = 2 - 8 = -6 \]
- For \( \begin{vmatrix} x & 1 \\ 8 & 1 \end{vmatrix} \):
\[ = (x \cdot 1) - (1 \cdot 8) = x - 8 \]
- For \( \begin{vmatrix} x & 2 \\ 8 & 8 \end{vmatrix} \):
\[ = (x \cdot 8) - (2 \cdot 8) = 8x - 16 \]

3. **Substituting back into the determinant**:
Now substituting these back into the determinant equation:
\[ 3(-6) + 2(x - 8) + (8x - 16) = 0 \]
Simplifying this:
\[ -18 + 2x - 16 + 8x - 16 = 0 \]
Combine like terms:
\[ 10x - 50 = 0 \]

4. **Solve for \( x \)**:
Now, we solve for \( x \):
\[ 10x = 50 \implies x = \frac{50}{10} = 5 \]

### Final Answer:
The value of \( x \) is \( 5 \).

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