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The general equation of a conic is $A x^2 + B x y + C y^2 + D x + E y + F = 0$. At Wikipedia, there is an equation for the eccentricity, based on ABCDEF.

Is there a similar equation for getting the foci or directrix for a general ellipse, parabola, hyperbola from ABCDEF? Please assume that a non-degenerate form of the equation is given.

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1 Answer 1

You'll first want to check either the discriminant or the eccentricity of your conic before proceeding to use any expression(s) for the foci; the central conics have two foci while the parabola only has one.

For the central conics, it is known that the two foci are at a distance $a\epsilon$ from the center, where $\epsilon$ is the eccentricity and $a$ is the semimajor axis for an ellipse, and the semitransverse axis for a hyperbola.

To simplify things, you'd first want to perform a translation on your central conic, such that the conic's center is now at the origin, and the standard equation becomes

$$\alpha x^2+\beta xy+\gamma y^2=1$$

with that, you can use the formula

$$a=\sqrt{\frac2{\alpha+\gamma+\mathrm{sgn}(\alpha-\gamma)\sqrt{\alpha^2+\beta^2+\gamma^2-2\alpha\gamma}}}$$

along with the eccentricity formula (like the one here) and the formula for the slope of the major/transverse axis to figure out the coordinates of your foci.

The parabolic case is a bit tricky, and I'll leave that to another answerer.

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