Sure! Here is the test case generator for the given problem statement:

```python
import random

def tcgen():
    N = random.randint(2, 10**4)
    coordinates = [(random.randint(1, 10**9), random.randint(1, 10**9)) for _ in range(N)]
    
    return N, coordinates

# Function to generate a formatted test case as a string
def generate_test_case():
    N, coordinates = tcgen()
    test_case = f"{N}\n"
    for x, y in coordinates:
        test_case += f"{x} {y}\n"
    return test_case

# Example usage
if __name__ == "__main__":
    print(generate_test_case())
```

This generator function `tcgen()` will create a random test case for the problem statement. It will generate:
- An integer \(N\) within the range [2, 10^4].
- A list of \(N\) tuples where each tuple contains two integers \(x_i\) and \(y_i\) within the range [1, 10^9].

The `generate_test_case()` function formats the output as a string similar to the input format required by the problem statement, which is useful for direct inclusion in test files or for competitive programming platforms.