Factorize ${(a x + b y)}^{2} + {(b x - a y)}^{2}$ by the grouping method

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

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

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

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

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Solution

The correct option is C $(x^{2} + y^{2}) (a^{2} + b^{2})$
${(a x + b y)}^{2} + {(b x - a y)}^{2}$
$= [(a x)^{2} + 2 (a x) (b y) + (b y)^{2}] + [(b x)^{2} - 2 (b x) (a y) + (a y)^{2}]$
$= a^{2} x^{2} + 2 a b x y + b^{2} y^{2} + b^{2} x^{2} - 2 a b x y + a^{2} y^{2}$
$= a^{2} x^{2} + a^{2} y^{2} + b^{2} y^{2} + b^{2} x^{2}$
$= a^{2} (x^{2} + y^{2}) + b^{2} (x^{2} + y^{2})$
$= (x^{2} + y^{2}) (a^{2} + b^{2})$

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