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Thread: direct variation

  1. #1
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    direct variation

    given
    $\displaystyle x^2 - y^2 \varpropto x^2 + y ^2$
    prove that
    (a)
    $\displaystyle y\varpropto x $
    (b)
    $\displaystyle x-y \varpropto x+ y$
    ------------------------------------------------------
    (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)
    $\displaystyle x^2 - y^2 = k (x^2 + y^2)$
    $\displaystyle (x-y)(x+y) = k ((x+y)^2 - 2xy)$
    $\displaystyle x-y =k(x+y - \frac {2xy}{x+y})$
    ???????????
    ??????????????????

    Thanks in advance.
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  2. #2
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    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).
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  3. #3
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    $\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
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  4. #4
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    anybody please help?
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  5. #5
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    $\displaystyle x+y = \sqrt\frac{1+k}{1-k}y + y $

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

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