# direct variation

#### cakeboby

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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#### sa-ri-ga-ma

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).

#### cakeboby

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

Cannot get it

#### cakeboby

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