Question

$\{\left(\frac{1}{x}\right)+\left(\frac{1}{y}\right)\}\infty \left\{\frac{1}{x+y}\right\}$, then prove $x\infty y$

Answer 1

Given, $\frac{1}{x}+\frac{1}{y}=\frac{k}{x+y}$
$\Rightarrow \left[\frac{(x+y)}{xy}\right]=\frac{k}{(x+y)}$
$\Rightarrow (x+y{)}^2=kxy$
$\Rightarrow x^2+y^2+2xy=kxy$

$\Rightarrow x^2+y^2=xy(k-2)$
$\Rightarrow \frac{x+y^2}{x}=y(k-2)$
$\Rightarrow x=y(k-2)-y^2/x$
$\Rightarrow x=y\left[\right(k-2)-y/x]$
$\Rightarrow x=yG$
So, we can say $x$ is directly proportional to $y$.