Reduced equation of the horizontal hyperbola
Find the equation of the hyperbola with center in the coordinated origin, focal distance $c=4$ and eccentricity $e\geq 1$ to be chosen.
$e=4$ is chosen. With $e=\dfrac{c}{a} \Rightarrow a=\dfrac{c}{e}=\dfrac{4}{4}=1$.
As $c^2=a^2+b^2 \Rightarrow b=\sqrt{c^2-a^2}=\sqrt{16-1}=\sqrt{15}$.
Substituting in $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ the equation is obtained $$\dfrac{x^2}{1}-\dfrac{y^2}{15}=1$$
For $e=4$, the equation is $\dfrac{x^2}{1}-\dfrac{y^2}{15}=1$.