Question

p and q are positive integers and \[ \frac{p}{q} + \frac{q}{p} = 3, \] then, \[ \frac{p^2}{q^2} + \frac{q^2}{p^2} = \]

• A. 3
• B. 7
• C. 9
• D. 11

Hint:

When given a sum of fractions like \( \frac{p}{q} + \frac{q}{p} \), square the equation to simplify and find the desired expression.

Answer 1

The Correct Option is B

We are given that \( \frac{p}{q} + \frac{q}{p} = 3 \). Let us square both sides of this equation: \[ \left( \frac{p}{q} + \frac{q}{p} \right)^2 = 3^2 \] Expanding the left-hand side: \[ \frac{p^2}{q^2} + 2 + \frac{q^2}{p^2} = 9 \] \[ \frac{p^2}{q^2} + \frac{q^2}{p^2} = 9 - 2 = 7 \] Thus, \( \frac{p^2}{q^2} + \frac{q^2}{p^2} = 7 \), so the correct answer is option (B).
Final Answer: (B) 7