The general equation of second degree` ax^(2) +2bxy +by^(2) +2yx +2fy +c = 0 ` represents a parabola if `Delta =abc +2fgh - af^(2) - bg^(2) - ch ^(2) ne 0 ` and _________

Answer

To determine the conditions under which the general equation of the second degree \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) represents a parabola, we need to analyze the discriminant and the coefficients involved.

### Step-by-Step Solution:

1. **Identify the General Form**:
The general equation of the second degree is given as:
\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]

2. **Discriminant Condition**:
For the conic section represented by this equation to be a parabola, we need to satisfy the condition:
\[ \Delta = abc + 2fgh - af^2 - bg^2 - ch^2 \neq 0 \]

3. **Condition on Coefficients**:
Additionally, there is another condition that must be satisfied:
\[ h^2 - ab = 0 \]
This implies that \( h^2 = ab \).

4. **Interpretation of Conditions**:
- The first condition ensures that the equation does not degenerate into a line or a point.
- The second condition \( h^2 - ab = 0 \) indicates that the conic is a parabola, as it reflects the relationship between the coefficients of the quadratic terms.

5. **Conclusion**:
Therefore, the general equation represents a parabola if:
\[ \Delta \neq 0 \quad \text{and} \quad h^2 - ab = 0 \]

### Final Answer:
The general equation represents a parabola if:
\[ \Delta = abc + 2fgh - af^2 - bg^2 - ch^2 \neq 0 \quad \text{and} \quad h^2 - ab = 0 \]