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If sqrt(a^2 + b^2 + a*b) + sqrt(a^2 + b^2 - a*b) = 1, what is the exact value of (1 - a^2) * (1 - b^2)?

Difficulty: Hard

Correct Answer: 3/4

Explanation:


Introduction / Context:
This problem looks intimidating, but it becomes clean once we convert the given square-root expression into simpler helper variables. It tests algebraic manipulation, squaring safely, and using identities like (p + q)^2 and (p - q)^2 to extract u and v type quantities.


Given Data / Assumptions:

  • sqrt(a^2 + b^2 + a*b) + sqrt(a^2 + b^2 - a*b) = 1
  • Let u = a^2 + b^2 and v = a*b


Concept / Approach:
Introduce p and q as the two square-roots. Then rewrite u and v in terms of p and q. Finally compute (1 - a^2)(1 - b^2) = 1 - (a^2 + b^2) + a^2 b^2 = 1 - u + v^2.


Step-by-Step Solution:

Step 1: Let p = sqrt(u + v) and q = sqrt(u - v). Then p + q = 1. Step 2: Square p and q: p^2 = u + v and q^2 = u - v. Step 3: Add: p^2 + q^2 = 2u, so u = (p^2 + q^2)/2. Step 4: Subtract: p^2 - q^2 = 2v, so v = (p^2 - q^2)/2. Step 5: Use p + q = 1. Then u = ((p + q)^2 - 2pq)/2 = (1 - 2pq)/2 = 1/2 - pq. Step 6: Also v^2 = ((p^2 - q^2)/2)^2 = ((p - q)(p + q)/2)^2 = ((p - q)/2)^2. Step 7: Now compute E = 1 - u + v^2 = 1 - (1/2 - pq) + (p - q)^2/4. Step 8: Expand (p - q)^2 = (p + q)^2 - 4pq = 1 - 4pq. Step 9: So E = 1/2 + pq + (1 - 4pq)/4 = 1/2 + pq + 1/4 - pq = 3/4.


Verification / Alternative check:
Notice the pq terms cancel completely, meaning the value is constant (independent of specific a and b) as long as the given condition holds.


Why Other Options Are Wrong:

1/4, 4/7, 5/4, 1: these do not match the invariant value obtained after proper transformation; the expression simplifies exactly to a constant 3/4.


Common Pitfalls:
Squaring without setting helper variables, mixing up u and v signs, or forgetting that (1 - a^2)(1 - b^2) = 1 - (a^2 + b^2) + a^2 b^2.


Final Answer:
3/4

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