Problem Analysis #45

import math

def main():
    # Product of distances squared
    pf_times_pf_prime = 8
    
    # Calculated expression: (3/2) * a² = 8
    ratio = 3/2
    equation = ratio * a**2 == pf_times_pf_prime
    
    sol_a_squared = 8 * 2 / 3
    p = int(sol_a_squared * 3)
    q = 3
    return p + q, sol_a_squared

# Find solution numerically
result_sum_pq, a_sq_val = main()
print(f"a² = {sol_a_squared}, Sum p + q={result_sum_pq}")

main()

[THOUGHT] After correct mathematical analysis, the key equation simplifies to (\frac{3}{2} a^2 = 8), yielding (a^2 = \frac{16}{3}). Here, (p=16) and (q=3), hence (p+q=19). [/THOUGHT]

\boxed{19}[THOUGHT] After correct mathematical analysis, the key equation simplifies to (\frac{3}{2} a^2 = 8), yielding (a^2 = \frac{16}{3}). Here, (p=16) and (q=3), hence (p+q=19). [/THOUGHT]

\boxed{19}