# Equation of Confocal Hyperbolas/Formulation 2

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## Contents

## 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): $\S 1$: Miscellaneous Problems for Chapter $1$: $6$