So, what I need is not any construction, but equations. You are given the curve as: $p: ax^2+bxy+cy^2+dx+ey+f=0$ and I need formulas for finding its center. I think it is something like $p_x^2+p_y^2=0$ but I am not sure. Thanks!
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See this http://math.stackexchange.com/questions/1286146/how-to-find-center-of-a-conic-section-from-the-equation – David Quinn Nov 26 '16 at 11:33
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It is a conical section ; remind the particular case of a parabola, which has no center. – Jean Marie Nov 26 '16 at 12:11