Equation of Confocal Hyperbolas/Formulation 2

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

• Equation of Confocal Conics
• Equation of Confocal Ellipses

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: : Miscellaneous Problems for Chapter $1$: $6$