Identify the surface defined by the following equation.

$x^2+y^2+7z^2+8x=-15$

Select the correct choice below and fill in the answer boxes to complete your choice.

A$.$ The surface defined by the equation is a hyperboloid of one sheet centered at $($ and is centered around the z$-$axis. In the xy$-$plane, the trace has length $$ in the $x-$direction and length $$ in the $y-$direction.

B$.$ The surface defined by the equation is an ellipsoid centered at $,$ that has length $$ in the $x-$direction, $$ in the $y-$direction, and $$ in the $z-$direction.

C$.$ The surface defined by the equation is an elliptic cone centered at $$ and whose traces in planes parallel to the $xy-$plane are ellipses.

D$.$ The surface defined by the equation is a hyperboloid of two sheets centered at $|$ and whose traces in planes parallel to the xy$-$plane are ellipses. The shortest distance between the two sheets is $$