Question:
Factorise:
(ax + by)2 + (bx − ay)2
Solution:
We have:
$(a x+b y)^{2}+(b x-a y)^{2}=\left(a^{2} x^{2}+b^{2} y^{2}+2 a x b y\right)+\left(b^{2} x^{2}+a^{2} y^{2}-2 b x a y\right)$
$=a^{2} x^{2}+a^{2} y^{2}+b^{2} y^{2}+b^{2} x^{2}+2 a x b y-2 b x a y$
$=a^{2}\left(x^{2}+y^{2}\right)+b^{2} x^{2}+b^{2} y^{2}+2 a x b y-2 a x b y$
$=a^{2}\left(x^{2}+y^{2}\right)+b^{2}\left(x^{2}+y^{2}\right)$
$=\left(x^{2}+y^{2}\right)\left(a^{2}+b^{2}\right)$
$\therefore(a x+b y)^{2}+(b x-a y)^{2}=\left(x^{2}+y^{2}\right)\left(a^{2}+b^{2}\right)$
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