Question

if
$\frac{1}{x} + \frac{1}{y} \infty \frac{1}{x + y}$
then prove that
$x \infty y$
​

Answer 1

$\rm\frac{1}{y} -\frac{1}{x} \propto \frac{1}{x-y}\\\ \Rightarrow \frac{1}{y}-\frac{1}{x}=\frac{k}{x-y},\,k \text{ being the constant of variation.}\\\Rightarrow\frac{x-y}{xy}=\frac{k}{x-y}\\\Rightarrow (x-y)^2 = kxy\\\Rightarrow x^2 -2xy+y^2 =kxy\\\Rightarrow x^2 -(k+2)xy+y^2 =0\\\Rightarrow x^2 -mxy +y^2 =0,\text{ where }m=k+2\text{ is a}\\\text{constant.}\\\Rightarrow \left(\frac{x}{y}\right)^2 -m\left(\frac{x}{y}\right)+1=0$

$[\text{on dividing both the sides by }y^2.]$

$\Rightarrow \frac{x}{y} = \frac{m\pm \sqrt{m^2 -4}}{2}=l\,(\text{say}),\text{a constant.}$

$[\text{by quadratic formula.}]$

$\Rightarrow x =ly$

This suggests that:

$\boxed{x \propto y.}$ $\dagger$

Hope,this will help you!!