Equation of Confocal Ellipses/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 such that $a > c$

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

Proof
Let $a$ and $c$ be arbitrary (strictly) positive real numbers fulfilling the constraints as defined.

Let $E$ be the locus of the equation:
 * $(1): \quad \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$

As $a > c$ it follows that:
 * $a^2 > c^2$

and so:
 * $a^2 - c^2 > 0$

Thus $(1)$ is in the form:
 * $\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$

From Equation of Ellipse in Reduced Form, this is the equation of an ellipse in reduced form.

Thus:
 * $\tuple {\pm a, 0}$ are the positions of the vertices of $E$
 * $\tuple {0, \pm b}$ are the positions of the covertices of $E$
 * $\tuple {\pm c, 0}$ are the positions of the foci of $E$.

Hence the result.

Also see

 * Equation of Confocal Conics
 * Equation of Confocal Hyperbolas