Equation of Confocal Conics/Formulation 1/Also presented as
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Equation of Confocal Conics: Formulation $1$: Also presented as
The equation for confocal conics can also be seen presented in the form:
- $\dfrac {x^2} {a^2 - k} + \dfrac {y^2} {b^2 - k} = 1$
where:
- $\tuple {x, y}$ denotes an arbitrary point in the cartesian plane
- $a$ and $b$ are real constants such that $a^2 > b^2$
- $k$ is a real parameter such that $a^2 > k$.
In this presentation:
- if $b^2 > k$, the equation generates confocal ellipses whose foci are at $\tuple {\pm \sqrt {a^2 - b^2}, 0}$
- if $b^2 < k < a^2$, the equation generates confocal hyperbolas whose foci are at $\tuple {\pm \sqrt {a^2 - b^2}, 0}$.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): confocal conics
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): confocal conics