Here's the test case generator for the provided problem statement:

```python
import random

def tcgen():
    N = random.randint(2, 2 * 10**5)
    
    points = [(random.randint(1, 10**9), random.randint(1, 10**9)) for _ in range(N)]
    
    return N, points

# Example of generating and printing a test case
def print_test_case():
    N, points = tcgen()
    print(N)
    for x, y in points:
        print(x, y)

# Generate and print a test case
print_test_case()
```

This function `tcgen` generates a random test case for the problem. It selects a random number \( N \) of points within the given constraints, then generates \( N \) points with coordinates within the range [1, \( 10^9 \)]. The `print_test_case` function prints the generated test case in the required format.

To use this generator, simply call `print_test_case`, and it will print a randomly generated test case to the standard output. You can modify this function to store test cases in a file or use them in any other way as per your needs.