# Thread: direct variation

1. ## direct variation

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

2. Add y on both side and subtract y on both side and find (x+y) and (x-y) and take the ratio of (x+y)/(x-y).

3. $\displaystyle x = \sqrt {\frac {k+1}{1-k}}y$
$\displaystyle x+y = \sqrt {\frac {k+1}{1-k}}y+y$
$\displaystyle x+y =( \sqrt {\frac {k+1}{1-k}}+1)y$

Cannot get it

5. $\displaystyle x+y = \sqrt\frac{1+k}{1-k}y + y$
$\displaystyle x-y = \sqrt\frac{1+k}{1-k}y - y$
$\displaystyle \frac{(x+y)}{(x-y)} = \frac{\sqrt\frac{1+k}{1-k} + 1 }{\sqrt\frac{1+k}{1-k} - 1}$