Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

3 Answers 3

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.

share|improve this answer
    
@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)

share|improve this answer

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.