Question

If $\left(\frac{x}{y}\right)=\left(\frac{z}{w}\right)$, then what is ${(xy+zw)}^2$ equal to ?

• A. $(x^2+z^2)(y^2+w^2)$
• B. $x^2{y}^2+z^2{w}^2$
• C. $x^2{w}^2+y^2{z}^2$
• D. $(x^2+w^2)(y^2+z^2)$

Answer 1

The correct option is A $(x^2+z^2)(y^2+w^2)$

$(xy+zw{)}^2=x^2{y}^2+z^2{w}^2+2xyzw$

As $\frac{x}{y}=\frac{z}{w}\Rightarrow xw=yz$

$\therefore (xy+zw{)}^2=x^2{y}^2{z}^2{w}^2+(xw\left)\right(yz)+(xw\left)\right(yz)$

$=x^2{y}^2+z^2{w}^2+y^2{z}^2+x^2{w}^2$

$\Rightarrow y^2(x^2+z^2)+w^2(x^2+z^2)$

$=(x^2+z^2)(y^2+w^2)$