use the Parameterization in u and v to write the term $x^2+y^2$

Question

Given that :

$u=xy$

$v=x^2-y^2$

we want to write the term $x^2+y^2 $ using only $u$ and $v$.

how can we do this ?

update: please reread my question I have edited it. I think it is clear now sorry for misconception

Answer 1

Suppose $u = xy$, $v = x^2 - y^2$

Then $$x^2 = \frac{u^2}{y^2} = v + y^2$$, so $$y^4 + vy^2 - u^2 = 0$$

Let $z = y^2$, then $$z^2 + vz - u^2 = 0 \implies y^2 = z = \frac{-v \pm \sqrt{v^2 - 4u^2}}{2}$$

Similarly, $$y^2 = \frac{u^2}{x^2} = x^2 - v$$ so $$x^4 - vx^2 - u^2 = 0$$ Let $w = x^2$, then $$x^4 - vx^2 - u^2 = 0 \implies x^2 = w = \frac{v \pm \sqrt{v^2 + 4u^2}}{2}$$

So $$x^2 + y^2 = \frac12(\pm \sqrt{v^2 - 4u^2} \pm \sqrt{v^2 + 4u^2})$$

I think I may have added some extraneous solutions. Either that or there is no convenient singular solution. Of course, the sign of $v^2 - 4u^2$ needs to be taken into account.