Sure, here is a test case generator for the given problem:

```python
import random

def generate_coordinates(N):
    coordinates = set()
    while len(coordinates) < N:
        x = random.randint(-10**9, 10**9)
        y = random.randint(-10**9, 10**9)
        coordinates.add((x, y))
    return list(coordinates)

def tcgen():
    N = random.randint(1, 50)
    coordinates = generate_coordinates(N)
    
    return N, coordinates

# Example usage
if __name__ == "__main__":
    N, coordinates = tcgen()
    print(N)
    for x, y in coordinates:
        print(x, y)
```

### Explanation:

- **generate_coordinates(N)**: This function generates a set of \(N\) unique coordinate pairs \((x, y)\) where \(|x|\) and \(|y|\) are within the range \([-10^9, 10^9]\).
- **tcgen()**: This function generates a random integer \(N\) within the range \([1, 50]\) and then calls `generate_coordinates(N)` to get the unique coordinates.
- **Example usage**: In the example usage section, the `tcgen` function is called, and the generated number of balls \(N\) and their coordinates are printed in the required format.

You can run this code to generate random test cases for the given problem statement.