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$|\sqrt{(x-c)^2+y^2}-\sqrt{(x+c)^2+y^2}|=2\frac{c}{e}$

Is there a way to remove the roots from the first term, so that there are no roots at all in the equation? But without substituting anything.

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  • $\begingroup$ are you familiar with hyperbola? this is exact what you have typed. $\endgroup$ – TIWARI Jun 24 '15 at 4:48
  • $\begingroup$ @TIWARI yes I am. $\endgroup$ – Arthur Jun 24 '15 at 4:49
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You can square both sides, which removes the absolute value sign. That will leave you with the cross term, which is $-2 \sqrt {[(x-c)^2+y^2][(x+c)^2+y^2]}$. Put everything else on the other side and square again. The square roots will be gone. Whether this is the easiest way to get where you are going is for you to judge, but the roots are gone.

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you can make the analogous to hyperbola,

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=0$

then substitute $a=\frac{c}{e}$ and $b=\sqrt{(\frac{c}{e})^2(e^2-1)}$

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