The following lines are from Schinzels paper "integer points on conics"
Let $ax^2 + bxy + cy^2 + dx + ey +f = 0$ be a non-degenerate conic over integers.
Setting $X= \delta x+be-2cd$, $Y=\delta y+bd-2ae$, where $\delta=b^2-4ac$, we get $$aX^2 +bXY+cY^2=4\delta \Delta \ \ \ (2) $$ for some integer $\Delta$.
Let $(u_0,v_0)$ be the least positive solution for $u^2-4\delta^3v^2=4$ and set $\varepsilon=(\frac{u_0+2\delta \sqrt{\delta}v_0}{2})^2 $.
If $X_0,Y_0$ is a solution of (2) and satisfies the congruences $$ X_0 \equiv be-2cd \bmod \delta,\ \ \ Y_0 \equiv bd-2ae \bmod \delta$$ then also every solution $X,Y$ of $$ 2aX +(b+\sqrt{\delta})Y = (2aX_0 +(b+\sqrt{\delta})Y_0) \varepsilon ^n$$satisfies these congrunces. Therefore we can assume that $$ 4 \sqrt{|a\delta\Delta|} \leq |2aX_0 +(b+\sqrt{\delta})Y_0| \leq 4 \sqrt{|a\delta\Delta|}\varepsilon $$ Question: Can somebody explain me why we can assume this?