Question Statement:
For positive non-zero real variables p and q, if
log (p2 + q2) = log p + log q + 2 log 3 ,
then, the value of
p4+q4
p2q2 is
Options:
-
(A) 79
-
(B) 81
-
(C) 9
-
(D) 83
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2024 Q.5)**
**1. Governing Concept**:
- **Topic**: Logarithmic Identities & Algebraic Manipulation
- Given equation:
$\log(p^2 + q^2) = \log p + \log q + 2\log 3 = \log(p \cdot q \cdot 3^2) = \log(9pq)$
- Exponentiating both sides:
$p^2 + q^2 = 9pq$
- Squaring both sides:
$(p^2 + q^2)^2 = (9pq)^2 \implies p^4 + 2p^2q^2 + q^4 = 81p^2q^2$
$p^4 + q^4 = 81p^2q^2 - 2p^2q^2 = 79p^2q^2$
- Dividing by $p^2q^2$:
$\frac{p^4 + q^4}{p^2q^2} = 79$
**2. Option Analysis**:
- **Option (A) [79] is CORRECT**: Exactly matches $81 - 2 = 79$.
- **Option (B) [81] is INCORRECT**: Omits subtracting the cross-term $2p^2q^2$.
- **Option (C) [9] is INCORRECT**: Equals the un-squared factor $9$.
- **Option (D) [83] is INCORRECT**: Erroneously adds $2$ instead of subtracting ($81 + 2 = 83$).
**3. Final Answer**:
- **Correct Choice**: **A**
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