Determine the set of parametric equations for the hyperbola with vertices at $(5,0)$ and $(-5,0)$, and foci at $(8,0)$ and $(-8,0)$.

$x=5\sec \left(t\right),y=\sqrt{39}\tan \left(t\right)$

$x=\sqrt{39}\sec \left(t\right),y=5\tan \left(t\right)$

$x=8\sec \left(t\right),y=5\tan \left(t\right)$

$x=5\tan \left(t\right),y=\sqrt{39}\sec \left(t\right)$