If a,b are distinct and different from zero, then `(a^(2),a),(b^(2),b)` and (0,0) are collinear.

Answer

To determine whether the points \((a^2, a)\), \((b^2, b)\), and \((0, 0)\) are collinear, we can use the concept of the area of the triangle formed by these three points. If the area is zero, then the points are collinear. We can calculate this using the determinant method.

### Step-by-step Solution:

1. **Set up the points**: We have three points:
- \( P_1 = (a^2, a) \)
- \( P_2 = (b^2, b) \)
- \( P_3 = (0, 0) \)

2. **Form the determinant**: The area of the triangle formed by these points can be calculated using the following determinant:
\[ \Delta = \begin{vmatrix} a^2 & a & 1 \\ b^2 & b & 1 \\ 0 & 0 & 1 \end{vmatrix} \]

3. **Calculate the determinant**: We can expand this determinant along the third row:
\[ \Delta = 1 \cdot \begin{vmatrix} a^2 & a \\ b^2 & b \end{vmatrix} \]
The determinant of the \(2 \times 2\) matrix is calculated as follows:
\[ \begin{vmatrix} a^2 & a \\ b^2 & b \end{vmatrix} = a^2 \cdot b - b^2 \cdot a = ab - a^2b = a(b - a) \]

4. **Final expression for the determinant**: Thus, we have:
\[ \Delta = a(b - a) \]

5. **Analyze the result**: Since \(a\) and \(b\) are distinct and non-zero, \(b - a \neq 0\) and \(a \neq 0\). Therefore, \(\Delta \neq 0\).

6. **Conclusion**: Since the determinant is not equal to zero, the points \((a^2, a)\), \((b^2, b)\), and \((0, 0)\) are not collinear.

### Final Answer:
The points \((a^2, a)\), \((b^2, b)\), and \((0, 0)\) are not collinear.