\(\displaystyle x^2 - y^2 \varpropto x^2 + y ^2\)

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.