I need help with hyperbolas please? Posting the hyperbola equations and questions in the comments.

I need help with hyperbolas please? Posting the hyperbola equations and questions in the comments.

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x^2 - y^2 = 1

\[\frac{ (x-1)^{2} }{ 2 }-\frac{ (y+3)^{2} }{ 2 }=1\]

The center of the hyperbola is (1, -3), right?

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