# Equation of Hyperbola in Reduced Form/Cartesian Frame

## Theorem

Let $K$ be an hyperbola aligned in a cartesian coordinate plane in reduced form.

Let:

- the transverse axis of $K$ have length $2 a$
- the conjugate axis of $K$ have length $2 b$.

The equation of $K$ is:

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

## Proof

By definition, the foci $F_1$ and $F_2$ of $K$ are located at $\tuple {-c, 0}$ and $\tuple {c, 0}$ respectively.

Let the vertices of $K$ be $V_1$ and $V_2$.

By definition, these are located at $\tuple {-a, 0}$ and $\tuple {a, 0}$.

Let the covertices of $K$ be $C_1$ and $C_2$.

By definition, these are located at $\tuple {0, -b}$ and $\tuple {0, b}$.

Let $P = \tuple {x, y}$ be an arbitrary point on the locus of $K$.

From the equidistance property of $K$ we have that:

- $\size {F_1 P - F_2 P} = d$

where $d$ is a constant for this particular ellipse.

From Equidistance of Hyperbola equals Transverse Axis:

- $d = 2 a$

Also, from Focus of Hyperbola from Transverse and Conjugate Axis:

- $c^2 a^2 = b^2$

Without loss of generality, let us choose a point $P$ such that $F_1 P > F_2 P$.

Then:

\(\displaystyle \sqrt {\paren {x + c}^2 + y^2} - \sqrt {\paren {x - c}^2 + y^2}\) | \(=\) | \(\displaystyle d = 2 a\) | Pythagoras's Theorem | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \sqrt {\paren {x + c}^2 + y^2}\) | \(=\) | \(\displaystyle 2 a + \sqrt {\paren {x - c}^2 + y^2}\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {x + c}^2 + y^2\) | \(=\) | \(\displaystyle \paren {2 a + \sqrt {\paren {x - c}^2 + y^2} }^2\) | squaring both sides | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x^2 + 2 c x + c^2 + y^2\) | \(=\) | \(\displaystyle 4 a^2 + 4 a \sqrt {\paren {x - c}^2 + y^2} + \paren {x - c}^2 + y^2\) | expanding | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x^2 + 2 c x + c^2 + y^2\) | \(=\) | \(\displaystyle 4 a^2 + 4 a \sqrt {\paren {x - c}^2 + y^2} + x^2 - 2 c x + c^2 + y^2\) | further expanding | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle c x - a^2\) | \(=\) | \(\displaystyle a \sqrt {\paren {x - c}^2 + y^2}\) | gathering terms and simplifying | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {c x - a^2}^2\) | \(=\) | \(\displaystyle a^2 \paren {\paren {x - c}^2 + y^2}^2\) | squaring both sides | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle c^2 x^2 - 2 c x a^2 + a^4\) | \(=\) | \(\displaystyle a^2 x^2 - 2 c x a^2 + a^2 c^2 + a^2 y^2\) | expanding | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle c^2 x^2 + a^4\) | \(=\) | \(\displaystyle a^2 x^2 + a^2 c^2 + a^2 y^2\) | simplifying | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a^2 c^2 - a^4\) | \(=\) | \(\displaystyle c^2 x^2 - a^2 x^2 - a^2 y^2\) | gathering terms | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a^2 \paren {c^2 - a^2}\) | \(=\) | \(\displaystyle \paren {c^2 - a^2} x^2 - a^2 y^2\) | simplifying | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a^2 b^2\) | \(=\) | \(\displaystyle b^2 x^2 - a^2 y^2\) | substituting $c^2 - a^2 = b^2$ from $(2)$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 1\) | \(=\) | \(\displaystyle \frac {x^2} {a^2} - \frac {y^2} {b^2}\) | dividing by $a^2 b^2$ |

$\blacksquare$

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**hyperbola** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**hyperbola**

- Weisstein, Eric W. "Hyperbola." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/Hyperbola.html