In any triangle ABC, if `2Delta a-b^(2)c=c^(3)`, (where `Delta` is the area of triangle), then which of the following is possible ?

Options

A. B is obtuse
B. A is obtuse
C. C is obtuse
D. B is right angle

Explanation

To solve the problem, we need to analyze the given equation in the context of triangle properties. The equation provided is:

\[ 2\Delta a - b^2c = c^3 \]

where \( \Delta \) is the area of triangle ABC, \( a \), \( b \), and \( c \) are the sides of the triangle opposite to angles A, B, and C respectively.

### Step-by-Step Solution:

1. **Rewrite the Equation**:
Start with the equation:
\[ 2\Delta a - b^2c = c^3 \]
Rearranging gives:
\[ 2\Delta a = b^2c + c^3 \]

2. **Express Area in Terms of Sides**:
The area \( \Delta \) of triangle ABC can be expressed using the formula:
\[ \Delta = \frac{1}{2}ab\sin C \]
Substitute this into the equation:
\[ 2 \left(\frac{1}{2}ab\sin C\right) a = b^2c + c^3 \]
Simplifying gives:
\[ a^2b\sin C = b^2c + c^3 \]

3. **Rearranging the Equation**:
Rearranging the equation yields:
\[ a^2b\sin C - b^2c - c^3 = 0 \]

4. **Using the Law of Cosines**:
We can relate the sides and angles of the triangle using the Law of Cosines:
\[ c^2 = a^2 + b^2 - 2ab\cos C \]

5. **Analyzing the Condition**:
We want to determine the conditions under which this equation holds. Particularly, we need to analyze the implications of the derived equation on the angles of the triangle.

6. **Considering the Angles**:
From the derived equation, we can analyze the implications for angle A, B, and C. We will particularly look for conditions that imply obtuse angles.

7. **Finding the Conditions for Obtuse Angles**:
If \( a^2b\sin C = b^2c + c^3 \) holds true, we can analyze the possibility of angle A being obtuse. For angle A to be obtuse, \( \cos A < 0 \) which implies that angle A is greater than 90 degrees.

8. **Conclusion**:
After analyzing the conditions, we find that the only possibility that satisfies the equation while keeping triangle properties intact is when angle A is obtuse.

### Final Answer:
Thus, the answer is that angle A is obtuse.