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I am trying to identifying the region represented by the equation:

$$x^2-y^2=9$$

I know that if it was a sum then it would be a circle but since it is a difference, how do I go about determining what type of region it is and how do I graph such equation?

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Your equation can be rewritten as $\dfrac {x^2}{3^2}-\dfrac {y^2}{3^2}=1$

Equations of the form $\dfrac {(x-h)^2}{a^2}-\dfrac {(y-k)^2}{b^2}=1$ are hyperbolas centered at $(h, k)$.

This is the graph for your equation:

Notice how the vertices of the hyperbola are at $(\pm3, 0)$.

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Hint: if $x$ and $y$ satisfy $x^2-y^2=9$, then either $y=\sqrt{x^2-9}$ or $y=-\sqrt{x^2-9}$.

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