If the hyperbola `x^2-y^2=4` is rotated by `45^0` in the anticlockwise direction about its center keeping the axis intact, then the equation of the hyperbola is `x y=a^2,` where `a^2` is equal to___________

Explanation

To solve the problem, we need to find the value of \( a^2 \) when the hyperbola \( x^2 - y^2 = 4 \) is rotated by \( 45^\circ \) in an anticlockwise direction about its center.

### Step-by-Step Solution:

1. **Identify the Original Hyperbola**:
The given hyperbola is \( x^2 - y^2 = 4 \). This can be rewritten in standard form as:
\[ \frac{x^2}{4} - \frac{y^2}{4} = 1 \]
Here, \( a^2 = 4 \).

2. **Understand the Rotation**:
When a hyperbola is rotated by \( 45^\circ \), the new equation can be represented in the form \( xy = c^2 \). There is a known relationship between the parameters of the hyperbola before and after rotation.

3. **Use the Relationship**:
The relationship between the original hyperbola and the rotated hyperbola is given by:
\[ c^2 = \frac{a^2}{2} \]
where \( c^2 \) corresponds to the new hyperbola \( xy = c^2 \) and \( a^2 \) corresponds to the original hyperbola \( x^2 - y^2 = a^2 \).

4. **Substitute the Value of \( a^2 \)**:
From the original hyperbola, we have \( a^2 = 4 \). Now substituting this into the relationship:
\[ c^2 = \frac{4}{2} = 2 \]

5. **Conclusion**:
Therefore, the value of \( a^2 \) in the equation \( xy = a^2 \) after the rotation is:
\[ a^2 = 2 \]

### Final Answer:
The value of \( a^2 \) is \( 2 \).