# Equation of Confocal Ellipses/Formulation 2

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## Contents

## 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 $E$

From Focus of Ellipse from Major and Minor Axis:

- $\tuple {\pm c, 0}$ are the positions of the foci of $E$.

Hence the result.

$\blacksquare$

## Also see

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 1$: Miscellaneous Problems for Chapter $1$: $6$