Focus of Hyperbola from Transverse and Conjugate Axis

Theorem
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.

Then:
 * $c^2 = a^2 + b^2$

Proof

 * HyperbolaFocusTransConj.png

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 $P = \left({x, y}\right)$ be an arbitrary point on the locus of $K$.

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


 * $\left\lvert{F_1 P + F_2 P}\right\rvert = d$

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.

Thus:

Some weird magic happens, and then: