Question
Factorise the following by taking out the common factors:$(mx + ny)^2 + (nx - my)^2$
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Answer
$(m x+n y)^2+(n x-m y)^2 $
$=m^2 x^2+n^2 y^2+2 m n x y+n^2 x^2+m^2 y^2-2 m n x y $
$=m^2 x^2+n^2 y^2+n^2 x^2+m^2 y^2 $
$=m^2 x^2+n^2 x^2+m^2 y^2+n^2 y^2 $
$=x^2\left(m^2+n^2\right)+y^2\left(m^2+n^2\right)$
Here, the common factor is $\left(m^2+n^2\right)$.
Dividing throughout by $\left(m^2+n^2\right)$, we get
$\frac{x^2\left(m^2+n^2\right)}{\left(m^2+n^2\right)}+\frac{y^2\left(m^2+n^2\right)}{\left(m^2+n^2\right)} $
$=x^2+y^2 $
$\therefore(m x+n y)^2+(n x-m y)^2 $
$=\left(m^2+n^2\right)\left(x^2+y^2\right) .$
$=m^2 x^2+n^2 y^2+2 m n x y+n^2 x^2+m^2 y^2-2 m n x y $
$=m^2 x^2+n^2 y^2+n^2 x^2+m^2 y^2 $
$=m^2 x^2+n^2 x^2+m^2 y^2+n^2 y^2 $
$=x^2\left(m^2+n^2\right)+y^2\left(m^2+n^2\right)$
Here, the common factor is $\left(m^2+n^2\right)$.
Dividing throughout by $\left(m^2+n^2\right)$, we get
$\frac{x^2\left(m^2+n^2\right)}{\left(m^2+n^2\right)}+\frac{y^2\left(m^2+n^2\right)}{\left(m^2+n^2\right)} $
$=x^2+y^2 $
$\therefore(m x+n y)^2+(n x-m y)^2 $
$=\left(m^2+n^2\right)\left(x^2+y^2\right) .$
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