Equation of Confocal Ellipses/Formulation 2
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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): $1$: The Nature of Differential Equations: Miscellaneous Problems for Chapter $1$: $6$