direct variation

Dec 2009
17
0
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)
\(\displaystyle x^2 - y^2 = k (x^2 + y^2)\)
\(\displaystyle (x-y)(x+y) = k ((x+y)^2 - 2xy)\)
\(\displaystyle x-y =k(x+y - \frac {2xy}{x+y})\)
???????????
??????????????????

Thanks in advance.
 
Jun 2009
806
275


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).
 
Dec 2009
17
0
\(\displaystyle x = \sqrt {\frac {k+1}{1-k}}y \)
\(\displaystyle x+y = \sqrt {\frac {k+1}{1-k}}y+y \)
\(\displaystyle x+y =( \sqrt {\frac {k+1}{1-k}}+1)y\)

Cannot get it
 
Jun 2009
806
275
\(\displaystyle x+y = \sqrt\frac{1+k}{1-k}y + y \)

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

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