Convert $x^2 + y^2 = xy$ into the form $f(x) = y$ OR $f(y) = x$?

Question

How should I take all the "x" elements free from y , in order to form a "function of x" or a " function of y " ?

Is this even possible ?

Is x*y representing multiplication? What happens when you solve this as a quadratic in $x$? — Mark Bennet, Oct 22 '18 at 16:25

score 1 · Answer 1 · answered Oct 22 '18 at 16:26

1

Your equation is possible only iff $x=y=0$:

$$x^2-xy+y^2 =0\implies 2x^2-2xy+2y^2=0$$ so $$(x-y)^2+x^2+y^2=0\implies x=y=0$$

So if you take a function $f(x)=x$ then $f(y) =x$.

answered Oct 22 '18 at 16:26

nonuser

2

What if $x$ and $y$ are not necessarily real? – Carl Schildkraut Oct 22 '18 at 16:41
@CarlSchildkraut then we have a (new) problem. Do you have any idea how to solve it? – nonuser Oct 22 '18 at 17:08

Awe Kumar Jha · Answer 2 · 2018-10-23T16:41:17.140

0

$$x^2 + y^2 = xy$$ $$~y^2 - x.y +x^2 = 0$$ $$~y = \frac {1 ± i√3}{2} x$$ $$~y = f(x)$$

edited Oct 23 '18 at 16:41

answered Oct 22 '18 at 16:35

Awe Kumar Jha

2 Answers2