$(a x + b y)^{2} + (b x - a y)^{2}$

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Solution

$(a x + b y)^{2} + (b x - a y)^{2} = a^{2} x^{2} + 2 a b x y + b^{2} y^{2} + b^{2} x^{2} - 2 a b x y + a^{2} y^{2} = x^{2} (a^{2} + b^{2}) + y^{2} (a^{2} + b^{2}) = (x^{2} + y^{2}) (a^{2} + b^{2})$

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