If ` P((-1)/(2),3) , Q(-5,6) and R (-8,8) ` are given Point which of the following is correct?

Options

A. Points form a triangle having non-zero area
B. Points lie on a circle
C. Points lie on a straight line
D. None of these

Answer

To determine the correct option regarding the points P, Q, and R, we need to find out if these points are collinear or not. We can do this by calculating the area of triangle PQR using the formula for the area of a triangle given its vertices.

### Step-by-Step Solution:

1. **Identify the coordinates of the points**:
- Point P: \( P\left(-\frac{1}{2}, 3\right) \)
- Point Q: \( Q(-5, 6) \)
- Point R: \( R(-8, 8) \)

2. **Use the formula for the area of a triangle**:
The area \( A \) of triangle formed by points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

3. **Substitute the coordinates into the formula**:
- Here, \( x_1 = -\frac{1}{2}, y_1 = 3 \)
- \( x_2 = -5, y_2 = 6 \)
- \( x_3 = -8, y_3 = 8 \)

Plugging in these values:
\[ A = \frac{1}{2} \left| -\frac{1}{2}(6 - 8) + (-5)(8 - 3) + (-8)(3 - 6) \right| \]

4. **Calculate each term**:
- First term: \( -\frac{1}{2}(6 - 8) = -\frac{1}{2}(-2) = 1 \)
- Second term: \( -5(8 - 3) = -5 \times 5 = -25 \)
- Third term: \( -8(3 - 6) = -8 \times (-3) = 24 \)

5. **Combine the terms**:
\[ A = \frac{1}{2} \left| 1 - 25 + 24 \right| = \frac{1}{2} \left| 0 \right| = 0 \]

6. **Conclusion**:
Since the area of triangle PQR is 0, the points P, Q, and R are collinear. Therefore, they lie on a straight line.

### Final Answer:
The correct option is that the points P, Q, and R are collinear (option 3).