Focus of Hyperbola from Transverse and Conjugate Axis

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Let $K$ be a hyperbola whose transverse axis is $2 a$ and whose conjugate axis is $2 b$.

Let $c$ be the distance of the foci of $K$ from the center.


$c^2 = a^2 + b^2$




Let the foci of $K$ be $F_1$ and $F_2$.

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

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

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 hyperbola.

This is true for all points on $K$.

In particular, it holds true for $V_2$, for example.


\(\ds d\) \(=\) \(\ds F_1 V_2 - F_2 V_2\)
\(\ds \) \(=\) \(\ds \paren {c + a} - \paren {c - a}\)
\(\ds \) \(=\) \(\ds 2 a\)

\(\ds c\) \(=\) \(\ds \sqrt {a^2 + b^2}\)
\(\ds \leadsto \ \ \) \(\ds a^2 + b^2\) \(=\) \(\ds c^2\)