Equation of Confocal Hyperbolas/Formulation 2

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The equation:

$\dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$


$\tuple {x, y}$ 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 $\tuple {\pm c, 0}$.


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.


Also see