Finding $a^2$ and $b^2$ for a hyperbola, knowing only the foci and point on curve

Question

I already know the numerical answers for this question, but I'm wondering if there is a way to get the vertex, $a$, and the co-vertex, $b$:

Each person is one mile ($5280$ feet) away from each other and each person is on the focus of the hyperbola. So, $F_1= (2640, 0)$ and $F_2= (-2640,0)$. If a lightning strike hit $12122$ feet north of $F_1$, could we find the vertices and co-vertices to produce the equation of this hyperbola? Or, do we need more information?

Answer 1

Yes, we could. A hyperbolla is fully defined by its foci and a point on it. Refer to the figure. $$|LF_2|^2=|LF_1|^2+|F_1F_2|^2,$$ thus $|LF_2|=13222.$ By definition of hyperbola, $$2a=\big||LF_2|-|LF_1|\big|=1100.$$ Finally, $$b^2=c^2-a^2=2640^2-550^2=6667100.$$

Note: If the foci and a point on the hyperbola are given, $a$ and $b$ can be constructed geometrically, without calculation. I did so for $b.$