Equation of Confocal Conics/Formulation 2

Definition

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 $\tuple {\pm c, 0}$.

Proof

Let $a > c$.

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

Let $a < c$.

Then from Equation of Confocal Hyperbolas, $(1)$ defines the set of all confocal hyperbolas whose foci are at $\tuple {\pm c, 0}$.

Hence the result.

$\blacksquare$

Also see

• Equation of Confocal Conics/Formulation 1

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: Miscellaneous Problems for Chapter $1$: $6$