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Why is $x=y$ an axis of symmetry of the figure described by $xy=1/3$?

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up vote 6 down vote accepted

Because if you interchange $x$ and $y$ the equation does not change.

Note that in general if $(a,b)$ is a point, then the point $(b,a)$ is the reflection of $(a,b)$ in the line $x=y$. If you have a more complicated equation, like $x^3+y^3=xy+25$ with the same property of being unchanged when $x$ and $y$ are interchanged, we will have the same phenomenon. Note that in this case the point $(1,3)$ is on the curve. Since $x$ and $y$ appear symmetrically in the equation, it is automatic that $(3,1)$ is also on the curve.

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@ChrisEagle: Thank you, I had managed with a simple typo to make the whole answer meaningless. – André Nicolas Apr 2 '12 at 11:47
Thank you very much ! – Jurgen Apr 2 '12 at 14:58

It is obvious, when you draw a picture:

enter image description here

(from here)

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Using the change of variables (rotation) $x=\frac{v-u}{\sqrt{2}}$ and $y=\frac{v+u}{\sqrt{2}}$, which sends $x=y$ to $u=0$, we get $$ \frac13=xy=\frac{v^2-u^2}{2}\tag{1} $$ Equation $(1)$ becomes $$ v^2=u^2+\frac23\tag{2} $$ Since we get the same values of $v$ for $u$ and $-u$, we see that $u=0$ ($x=y$) is an axis of symmetry.

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