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By using the formulas;
(a + b)2 = a2 + b2 + 2ab and
(a – b)2 = a2 + b2 – 2ab
= (a2x2 + b2y2 + 2axby) + (b2x2 + a2y2 – 2bxay)
= a2x2 + a2y2 + b2y2 + b2x2 + 2axby – 2bxay
= a2(x2 + y2) + b2x2 + b2y2 + 2axby – 2axby
= a2(x2 + y2) + b2(x2 + y2)
= (x2 + y2)(a2 + b2)
So we get,
(ax + by)2 + (bx – ay)2 = (x2 + y2)(a2 + b2)
Hope it will help you.