If $x$ varies as $y$ , prove that $x^{2} + y^{2}$ varies as $x^{2} - y^{2}$ .

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Solution

Let $x = k y$ then,

$x^{2} + y^{2} = k^{2} y^{2} + y^{2} = y^{2} (k^{2} + 1)$

$x^{2} - y^{2} = k^{2} y^{2} - y^{2} = y^{2} (k^{2} - 1)$

So

$\frac{x^{2} + y^{2}}{x^{2} - y^{2}} = \frac{(k^{2} + 1)}{(k^{2} - 1)}$

$x^{2} + y^{2} = l (x^{2} - y^{2})$ where $l = \frac{(k^{2} + 1)}{(k^{2} - 1)}$ , a constant.

Hence proved.

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