Equation of Conic Section/Cartesian Form/Discriminant Form

Theorem

Let $K$ be a conic section embedded in a Cartesian plane with the general equation:

$a x^2 + 2 h x y + b y^2 + 2 g x + 2 f y + c = 0$

where $a, b, c, f, g, h \in \R$.

Then after translation of coordinate axes, $K$ can be described using the equation:

$a x^2 + 2 h x y + b y^2 - \dfrac \Delta {h^2 - a b} = 0$

where $\Delta$ is the discriminant of $K$:

$\delta = \begin {vmatrix} a & h & g \\ h & b & f \\ g & f & c \end {vmatrix}$

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conic (conic section)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conic (conic section)