(ax + by)² + (bx – ay)²

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)