If the points (- 1, 2, - 3), (4, a, 1) and (b, 8, 5) are collinear, then : (a, b)`equiv`

Options

A. (5, 5)
B. (9, 5)
C. (5, 9)
D. (- 5, 9)

Answer

To determine the values of \( a \) and \( b \) such that the points (-1, 2, -3), (4, a, 1), and (b, 8, 5) are collinear, we can use the concept of the scalar triple product. The scalar triple product of three vectors is zero if the vectors are coplanar (which is the case for collinear points).

### Step-by-Step Solution:

1. **Define the Points as Vectors**:
Let:
- Point A = (-1, 2, -3) → Vector \( \mathbf{OA} = -1 \mathbf{i} + 2 \mathbf{j} - 3 \mathbf{k} \)
- Point B = (4, a, 1) → Vector \( \mathbf{OB} = 4 \mathbf{i} + a \mathbf{j} + 1 \mathbf{k} \)
- Point C = (b, 8, 5) → Vector \( \mathbf{OC} = b \mathbf{i} + 8 \mathbf{j} + 5 \mathbf{k} \)

2. **Set Up the Scalar Triple Product**:
The scalar triple product can be represented as:
\[ \mathbf{OA} \cdot (\mathbf{OB} \times \mathbf{OC}) = 0 \]
This means we need to calculate the determinant of the matrix formed by these vectors.

3. **Construct the Determinant**:
The determinant can be set up as follows:
\[ \begin{vmatrix} -1 & 2 & -3 \\ 4 & a & 1 \\ b & 8 & 5 \end{vmatrix} = 0 \]

4. **Calculate the Determinant**:
Expanding the determinant, we have:
\[ -1 \cdot \begin{vmatrix} a & 1 \\ 8 & 5 \end{vmatrix} - 2 \cdot \begin{vmatrix} 4 & 1 \\ b & 5 \end{vmatrix} - 3 \cdot \begin{vmatrix} 4 & a \\ b & 8 \end{vmatrix} \]
Calculating each of the 2x2 determinants:
- \( \begin{vmatrix} a & 1 \\ 8 & 5 \end{vmatrix} = 5a - 8 \)
- \( \begin{vmatrix} 4 & 1 \\ b & 5 \end{vmatrix} = 20 - b \)
- \( \begin{vmatrix} 4 & a \\ b & 8 \end{vmatrix} = 32 - ab \)

Putting it all together:
\[ -1(5a - 8) - 2(20 - b) - 3(32 - ab) = 0 \]
Simplifying this gives:
\[ -5a + 8 - 40 + 2b - 96 + 3ab = 0 \]
Combining like terms:
\[ 3ab - 5a + 2b - 128 = 0 \]

5. **Rearranging the Equation**:
Rearranging gives us:
\[ 3ab - 5a + 2b = 128 \]

### Conclusion:
The equation \( 3ab - 5a + 2b = 128 \) relates \( a \) and \( b \). To find specific values, we can try different pairs of integers for \( a \) and \( b \) that satisfy this equation.