Given the general equation of a conic section ($A x^2 + B x y + C y^2 + D x + E y + F = 0$), is there an simple expression for a focal point of the conic in terms of $ABCDEF$? The question has been asked here before, but the supplied answers transform the equation into complex coordinates, or transform the conic so that it's centered at the origin. Since none of the answers actually answer the question, one might suspect that there is no such expression. Is this true?
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2Simple? No, although one can come up with closed-form expressions. See here, for instance for formulas of the basic parameters of an ellipse in terms of the general equation’s coefficients. The expressions for coordinates of the foci would be at least as complicated. – amd Apr 01 '20 at 06:44
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1See $#393$ of https://archive.org/details/elementsofcoordi00loneuoft – lab bhattacharjee Apr 01 '20 at 06:59
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1Duplicate? How to find centre,vertics,foci,focal radii,letus rectum… when exists of a general quadratic equation in x and y – Blue Sep 10 '20 at 21:08