Question

For positive non-zero real variables $p$ and $q$, if  $\log (p^2 + q^2) = \log p + \log q + 2\log 3$, then, the value of $\frac{p^4 + q^4}{p^2 q^2}$ is:

• A. \( 79 \)
• B. \( 81 \)
• C. \( 9 \)
• D. \( 83 \)

Hint:

Use logarithmic properties to simplify equations and express them in terms of multiplication or addition.

Answer 1

The Correct Option is A

Given the equation: \[ \log (p^2 + q^2) = \log p + \log q + 2\log 3 \] Rewrite using logarithm properties: \[ \log (p^2 + q^2) = \log (pq) + \log 9 \] This simplifies to: \[ p^2 + q^2 = 9pq \] Squaring both sides: \[ (p^2 + q^2)^2 = (9pq)^2 \] \[ p^4 + q^4 + 2p^2q^2 = 81 p^2 q^2 \] \[ p^4 + q^4 = 81 p^2 q^2 - 2 p^2 q^2 \] \[ p^4 + q^4 = 79 p^2 q^2 \] Dividing by \( p^2 q^2 \): \[ \frac{p^4 + q^4}{p^2 q^2} = 79 \] Final Answer: \[ \boxed{79} \]