<s>[INST] The intersection of a line and a hyperbola produces two points:
\[ \left( \frac{1}{\sqrt{2}}, \frac{\sqrt{3}-1}{2\sqrt{2}} \right), \left( \frac{1}{\sqrt{2}}, -\frac{\sqrt{3}+1}{2\sqrt{2}} \right) \]

Find the equation of the hyperbola on which the line crosses these two points.[/INST]