# Equation of Confocal Hyperbolas/Formulation 1

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## Definition

The equation:

- $(1): \quad \dfrac {x^2} {a^2 + \lambda} + \dfrac {y^2} {b^2 + \lambda} = 1$

where:

- $\tuple {x, y}$ denotes an arbitrary point in the cartesian plane
- $a$ and $b$ are (strictly) positive constants such that $a^2 > b^2$
- $\lambda$ is a (strictly) positive parameter such that $b^2 < -\lambda < a^2$

defines the set of all confocal hyperbolas whose foci are at $\tuple {\pm \sqrt {a^2 + b^2}, 0}$.

## Proof

Let $a$ and $b$ be arbitrary (strictly) positive real numbers fulfilling the constraints as defined.

Let $E$ be the locus of the equation:

- $(1): \quad \dfrac {x^2} {a^2 + \lambda} + \dfrac {y^2} {b^2 + \lambda} = 1$

As $b^2 < -\lambda$ it follows that:

- $b^2 + \lambda < 0$

and as $-\lambda < a^2$:

- $a^2 + \lambda > 0$

Thus $(1)$ is in the form:

- $\dfrac {x^2} {r^2} - \dfrac {y^2} {s^2} = 1$

where:

- $r^2 = a^2 + \lambda$
- $s^2 = -\lambda + b^2$

From Equation of Hyperbola in Reduced Form, this is the equation of an hyperbola in reduced form.

It follows that:

- $\tuple {\pm \sqrt {a^2 + \lambda}, 0}$ are the positions of the vertices of $E$
- $\tuple {0, \pm \sqrt {b^2 - \lambda} }$ are the positions of the covertices of $E$

From Focus of Hyperbola from Transverse and Conjugate Axis, the positions of the foci of $E$ are given by:

- $\paren {a^2 + \lambda} + \paren {b^2 - \lambda} = c^2$

where $\tuple {\pm c, 0}$ are the positions of the foci of $E$.

Thus we have:

\(\displaystyle c^2\) | \(=\) | \(\displaystyle \paren {a^2 + \lambda} + \paren {b^2 - \lambda}\) | Focus of Hyperbola from Transverse and Conjugate Axis | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle a^2 + b^2\) |

Hence the result.

$\blacksquare$