Derive the following expression:

Question

Given the function $(x_1^2+x_2^2)^2-x_1^2+x_2^2=0$, where $r^4=x_1^2-x_2^2$ and $r^2=x_1^2+x_2^2$ for $-1\leq r \leq 1$ show that

$x_1=\frac{1}{\sqrt{2}}r\sqrt{1+r^2}$ and $x_2= \pm \frac{1}{\sqrt{2}}\sqrt{1-r^2}$.

Any help is much appreciated!

Answer 1

Hint: We have $x_{1}^2 = \frac{r^4 + r^2}{2}$ and $x_{2}^2 = \frac{r^2 - r^4}{2}$