given

$\displaystyle x^2 - y^2 \varpropto x^2 + y ^2$

prove that

(a)

$\displaystyle y\varpropto x $

(b)

$\displaystyle x-y \varpropto x+ y$

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(a)

$\displaystyle x^2 - y^2 = k (x^2 + y^2)$

$\displaystyle (1-k)x^2 = (k+1)y^2$

$\displaystyle x = \sqrt{\frac{k+1}{1-k}}y$

$\displaystyle y\varpropto x $

(b)

$\displaystyle x^2 - y^2 = k (x^2 + y^2)$

$\displaystyle (x-y)(x+y) = k ((x+y)^2 - 2xy)$

$\displaystyle x-y =k(x+y - \frac {2xy}{x+y})$

???????????

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Thanks in advance.