Variable rationality

Algebra Level pending

Given that x x , y y , p p and q q are non-zero real numbers such that

x 2 + y 2 p 2 + q 2 = x y p q \dfrac{x^2+y^2}{p^2+q^2} = \dfrac{xy}{pq}

Find the ratio x q : y p xq : yp .

2:3 1:2 3:2 Not enough information 3:1 1:3 2:1 1:1

Viki Zeta by Viki Zeta , 19, India

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1 solution

Viki Zeta
Sep 6, 2016

x 2 + y 2 p 2 + q 2 = x y p q ⟹ p q ( x 2 + y 2 ) = x y ( p 2 + q 2 ) ⟹ p q x 2 + p q y 2 = x y p 2 + x y q 2 ⟹ p q x 2 − x y q 2 = x y p 2 − p q y 2 ⟹ x q ( p x − q y ) = p y ( p x − q y ) ⟹ x q = p y ⟹ x q y p = 1 1 ∴ xq:yp = 1 : 1 \dfrac{x^2+y^2}{p^2+q^2}=\dfrac{xy}{pq} \\ \implies pq(x^2+y^2) = xy(p^2+q^2) \\ \implies pqx^2 + pqy^2 = xyp^2 + xyq^2 \\ \implies pqx^2 - xyq^2 = xyp^2 - pqy^2 \\ \implies xq(px - qy) = py(px - qy) \\ \implies xq = py \\ \implies \dfrac{xq}{yp} = \dfrac{1}{1} \\ \therefore \fbox{ xq:yp = 1 : 1 }

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