If the point (x , y) , (-5, -2) and (3, -5) are collinear, prove that `3x+8y+31=0`.

Answer

To prove that the points (x, y), (-5, -2), and (3, -5) are collinear, we can use the area of the triangle formed by these points. If the area is zero, then the points are collinear.

### Step-by-Step Solution:

1. **Identify the Points**:
We have three points:
- Point A: (x, y)
- Point B: (-5, -2)
- Point C: (3, -5)

2. **Formula for Area of Triangle**:
The area \( A \) of a triangle formed by three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For our points, we can substitute:
- \( (x_1, y_1) = (x, y) \)
- \( (x_2, y_2) = (-5, -2) \)
- \( (x_3, y_3) = (3, -5) \)

3. **Substituting the Points into the Area Formula**:
Plugging in the coordinates into the area formula:
\[ A = \frac{1}{2} \left| x((-2) - (-5)) + (-5)((-5) - y) + 3(y - (-2)) \right| \]
Simplifying the expressions:
\[ A = \frac{1}{2} \left| x(3) + (-5)(-5 - y) + 3(y + 2) \right| \]
\[ A = \frac{1}{2} \left| 3x + 25 + 5y + 3y + 6 \right| \]
\[ A = \frac{1}{2} \left| 3x + 8y + 31 \right| \]

4. **Setting the Area to Zero**:
Since the points are collinear, the area must be zero:
\[ \frac{1}{2} \left| 3x + 8y + 31 \right| = 0 \]
This implies:
\[ 3x + 8y + 31 = 0 \]

5. **Conclusion**:
We have shown that if the points (x, y), (-5, -2), and (3, -5) are collinear, then \( 3x + 8y + 31 = 0 \) is indeed true.