Equation of Confocal Conics/Formulation 1

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

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.

$\blacksquare$

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

1926: E.L. Ince: Ordinary Differential Equations ... (previous) ... (next): Chapter $\text I$: Introductory: $\S 1.201$ The Differential Equation of a Family of Confocal Conics