# Equation of Confocal Conics/Formulation 1

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## Definition

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 $\tuple {\pm \sqrt {a^2 - b^2}, 0}$.

## Proof

Let $b^2 > -\lambda$.

Then from Equation of Confocal Ellipses: Formulation 1, $(1)$ defines the set of all confocal ellipses whose foci are at $\tuple {\pm \sqrt {a^2 - b^2}, 0}$.

Let $b^2 < -\lambda < a^2$.

Then from Equation of Confocal Hyperbolas: Formulation 1, $(1)$ defines the set of all confocal hyperbolas whose foci are at $\tuple {\pm \sqrt {a^2 - b^2}, 0}$.

Hence the result.

$\blacksquare$