\[ \left(\frac{1}{x} - \frac{1}{y}\right) \propto \frac{1}{x - y} \] i.e., \[ \left(\frac{1}{x} - \frac{1}{y}\right) = k \cdot \frac{1}{x - y} \quad \text{[where \(k\) is a non-zero constant]} \] \[ \Rightarrow \frac{y - x}{xy} = k \cdot \frac{1}{x - y} \] \[ \Rightarrow \frac{-(x - y)}{xy} = k \cdot \frac{1}{x - y} \] \[ \Rightarrow -(x - y)^2 = kxy \] \[ \Rightarrow (x - y)^2 = -kxy \] \[ \Rightarrow x^2 + y^2 - 2xy = -kxy \] \[ \Rightarrow x^2 + y^2 = 2xy - kxy \] \[ \Rightarrow x^2 + y^2 = (2 - k)xy \] \[ \Rightarrow \frac{x^2 + y^2}{xy} = (2 - k) = \text{constant} \] \[ \therefore \quad (x^2 + y^2) \propto xy \quad \text{(Proved)} \]