Determination of the Foci, Directrices, Axes and Eccentricities of A Conic Whose Equation is Given With Numerical Coefficients

The method given in Smith's Conic Sections (1910), p. 258, fails, like other methods, to correlate foci, directrices, eccentricities and axes , the following method, based on the equations there, shows how that can be done. If the conic is an ellipse, the real eccentricity goes with the real foci and the imaginary eccentricity with the imaginary foci, but if the conic is a hyperbola there is no such relation, as both values of the eccentricity are real.