Find the equation of the hyperbola with center in the coordinated origin, focal distance $$c=4$$ and eccentricity $$e\geq 1$$ to be chosen.
See development and solution
Development:
$$e=4$$ 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 $$$\dfrac{x^2}{1}-\dfrac{y^2}{15}=1$$$
Solution:
For $$e=4$$, the equation is $$\dfrac{x^2}{1}-\dfrac{y^2}{15}=1$$.