where is my mistake with this equation ot hyperbolic problem? Let parabola  $\Gamma_{1}:$$y^2=4x$,and hyperbolic curve $\Gamma_{2}$: $x^2-y^2=1$.it is well known this two is symmetric.so  the two point  $A$ and $B$ about $x$ axial symmetry,or mean $x_{A}=x_{B}$.see figure,

but we use
$$\begin{cases}
y^2=4x\\
x^2-y^2=1
\end{cases}$$
we have
$$x^2-4x-1=0\Longrightarrow x=2+\sqrt{5},2-\sqrt{5}$$ so the two point $\color{red}{x_{A}=2+\sqrt{5},x_{B}=2-\sqrt{5}}?$,then 
contradiction it is well known$x_{A}=x_{B}$,where is mistake?and the second root $\color{blue}{x=2-\sqrt{5}}$ where is from?or where is point to this figure
Thanks help

The confusion arises from ignoring an implicit condition of the first equation, $y^2 = 4x$. Since $y^2 \geq 0$, we have the condition $x \geq 0$.
Thus of the two solutions for $x$ you have, the positive $x = 2 + \sqrt{5}$ is the only solution.
We see that there are two values of $y$ that satisfy the equations with the $x$ value above, and this agrees with the graphical representation.
