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

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Solution

Given, $(a x + b y) ² + (b x - a y) ²$

$= a ² x ² + b ² y ² + 2 a b x y + b ² x ² + a ² y ² - 2 a b x y$ $[∵ (a + b)^{2} = a^{2} + b^{2} + 2 a b, (a - b)^{2} = a^{2} + b^{2} - 2 a b]$

$= a ² x ² + b ² x ² + b ² y ² + a ² y ²$

$= x ² (a ² + b ²) + y ² (a ² + b ²)$

$= (a ² + b ²) (x ² + y ²)$

Hence, factorise of $(a x + b y)^{2} + (b x - a y)^{2}$ is $(a^{2} + b^{2}) (x^{2} + y^{2})$

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