If the points (-2,-5),(2,-2),(8,a) are collinear, then the value of a is ……….

Answer

To determine the value of \( a \) such that the points \((-2, -5)\), \((2, -2)\), and \((8, a)\) are collinear, we can use the concept of the area of a triangle formed by these three points. If the area is zero, then the points are collinear.

### Step-by-Step Solution:

1. **Set Up the Determinant**:
The area of the triangle formed by the points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the determinant:
\[ \text{Area} = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right| \]
For our points, we have:
\[ x_1 = -2, y_1 = -5 \\ x_2 = 2, y_2 = -2 \\ x_3 = 8, y_3 = a \]

2. **Construct the Determinant**:
Plugging in the values, we get:
\[ \begin{vmatrix} -2 & -5 & 1 \\ 2 & -2 & 1 \\ 8 & a & 1 \end{vmatrix} \]

3. **Calculate the Determinant**:
The determinant can be calculated as follows:
\[ = -2 \begin{vmatrix} -2 & 1 \\ a & 1 \end{vmatrix} - (-5) \begin{vmatrix} 2 & 1 \\ 8 & 1 \end{vmatrix} + 1 \begin{vmatrix} 2 & -2 \\ 8 & a \end{vmatrix} \]

Now, calculate each of the 2x2 determinants:
\[ = -2((-2)(1) - (a)(1)) + 5((2)(1) - (8)(1)) + ((2)(a) - (-2)(8)) \]
\[ = -2(-2 - a) + 5(2 - 8) + (2a + 16) \]
\[ = 4 + 2a - 30 + 2a + 16 \]
\[ = 4a - 10 \]

4. **Set the Determinant to Zero**:
For the points to be collinear, the area must be zero:
\[ 4a - 10 = 0 \]

5. **Solve for \( a \)**:
\[ 4a = 10 \\ a = \frac{10}{4} = \frac{5}{2} \]

### Final Answer:
The value of \( a \) is \( \frac{5}{2} \).

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