Equation of Confocal Ellipses

Theorem

Formulation 1

The equation:

$(1): \quad \dfrac {x^2} {a^2 + \lambda} + \dfrac {y^2} {b^2 + \lambda} = 1$

where:

$\tuple {x, y}$ denotes an arbitrary point in the cartesian plane

$a$ and $b$ are (strictly) positive constants such that $a^2 > b^2$

$\lambda$ is a (strictly) positive parameter such that $b^2 > -\lambda$

defines the set of all confocal ellipses whose foci are at $\tuple {\pm \sqrt {a^2 - b^2}, 0}$.

Formulation 2

The equation:

$(1): \quad \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$

where:

$\tuple {x, y}$ denotes an arbitrary point in the cartesian plane

$c$ is a (strictly) positive constant

$a$ is a (strictly) positive parameter such that $a > c$

defines the set of all confocal ellipses whose foci are at $\tuple {\pm c, 0}$.

Also see

• Equation of Confocal Conics
• Equation of Confocal Hyperbolas