Exercise 1148Linear algebra › Quadratic formsL1Added 3 Nov 2026

The hyperbola $x^2-y^2=1$

#hyperbola#quadratic form#geometry

Statement

Describe the set $H=\{(x,y): x^2-y^2=1\}$ .

Hint

The quadratic form $Q(x,y)=x^2-y^2$ has signature $(1,1,0)$ ; the set $Q=1$ is a hyperbola.

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Signature. $Q$ is indefinite with signature $(1,1,0)$ .

Geometry. $x^2-y^2=1$ is a hyperbola centered at the origin, with asymptotes $y=\pm x$ (given by $x^2-y^2=0$ ). The vertices are at $(\pm1,0)$ .

Conclusion. $H$ is a hyperbola with vertices $(\pm1,0)$ and asymptotes $y=\pm x$ .