$\begin{array}{l}(a^2 + b^2)(x^2 + y^2) = (ax + by)^2\\ \Leftrightarrow a^2x^2 + a^2y^2 + b^2x^2 + b^2y^2 = a^2x^2 + 2axby + b^2y^2\\ \Leftrightarrow a^2y^2 - 2aybx + b^2x^2 = 0\\ \Leftrightarrow (ay - bx)^2 = 0\\ \Leftrightarrow ay - bx = 0\\ \Leftrightarrow ay = bx\\ \Leftrightarrow \dfrac{a}{x} = \dfrac{b}{y} \qquad (\forall a,b,c,d \ne 0)\end{array}$