Equation of Confocal Hyperbolas/Formulation 2
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Definition
The equation:
- $\dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$
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 hyperbolas 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 $H$ 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:
- $\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$
Thus from Equation of Hyperbola in Reduced Form, $H$ defines an hyperbola where:
- $\tuple {\pm a, 0}$ are the positions of the vertices of $H$
- the transverse axis of $H$ has length $2 a$
- the conjugate axis of $H$ has length $2 b$
From Focus of Hyperbola from Transverse and Conjugate Axis
- $\tuple {\pm c, 0}$ are the positions of the foci of $H$.
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 $1$: $6$