What is factor (ax+by)^2+(ay-bx)^2 ?

The solution to factor (ax+by)^2+(ay-bx)^2 is (x^2+y^2)(a^2+b^2)

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factor (ax+by)^2+(ay-bx)^2

factor

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

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

Solution steps

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

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

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

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

= a^{2} x^{2} + 2 ab x y + b^{2} y^{2} + a^{2} y^{2} - 2 ab x y + b^{2} x^{2}

Simplify

a^{2} x^{2} + 2 ab x y + b^{2} y^{2} + a^{2} y^{2} - 2 ab x y + b^{2} x^{2} : a^{2} x^{2} + b^{2} x^{2} + b^{2} y^{2} + a^{2} y^{2}

= a^{2} x^{2} + b^{2} x^{2} + b^{2} y^{2} + a^{2} y^{2}

Factor out common term

a^{2}

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

Factor out common term

b^{2}

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

Factor out common term

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

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

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What is factor (ax+by)^2+(ay-bx)^2 ?
The solution to factor (ax+by)^2+(ay-bx)^2 is (x^2+y^2)(a^2+b^2)