Question

If $p=\frac{4xy}{x+y}$
Prove that :
$\frac{p+2x}{p-2x}+\frac{p+2y}{p-2y}=2$

Answer 1

$P=\frac{4xy}{x+y}$

$\frac{p+2x}{p-2x}+\frac{p+2y}{p-2y}$

$\Rightarrow \frac{\frac{4xy}{x+y}+2x}{\frac{4xy}{x+y}-2x}+\frac{\frac{4xy}{x+y}+2y}{\frac{4xy}{x+y}-2y}$

$\Rightarrow \frac{2x^2+6xy}{2xy-2x^2}+\frac{2y^2+6xy}{2xy-2y^2}$

$\Rightarrow \frac{x^2+3xy}{xy-x^2}+\frac{y^2+3xy}{xy-y^2}$

$\Rightarrow \frac{x+3y}{y-x}+\frac{y+3x}{x-y}$

$\Rightarrow \frac{x+3y}{y-x}-\frac{y+3x}{y-x}$

$\Rightarrow \frac{x+3y-y-3x}{y-x}=2$

$\Rightarrow \frac{2y-2x}{y-x}=2$

Hence proved