Equation of Conic Section/Cartesian Form/Eccentricity

Theorem

Let $K$ be a conic section embedded in a Cartesian plane such that:

one focus of $K$ is at the origin

the eccentricity of $K$ is $e$

the directrix of $K$ is a distance $h$ from the origin.

Then $K$ can be described using the equation:

$\paren {1 - e^2} x^2 + 2 e^2 h x + y^2 = e^2 h^2$

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)