Equation of Confocal Conics

Theorem

Formulation 1

The equation:

$\dfrac {x^2} {a^2 + \lambda} + \dfrac {y^2} {b^2 + \lambda} = 1$

where:

$\tuple {x, y}$ denotes an arbitrary point in the cartesian plane
$a$ and $b$ are real constants such that $a^2 > b^2$
$\lambda$ is a real parameter such that $a^2 > -\lambda$

defines the set of all confocal conics 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

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