Can I cut a hole in a 2" x 8" joist that is less than 1/3 the height of the joist, leaves at least 2" above and below the hole, but is 10" wide?
A wooden beam like used for a joist will typically face bending loads. You'll have a great deal of compressive stress on the top of the beam and a great deal of tension stress on the bottom beam. The center serves primarily to keep the top and bottom of the beam aligned so they act as a single beam, and is under shear stress parallel to the grain. 

As such, a hole like you pictured (in the center vertically and horizontally) will probably be fine. 

I'd be leery of doing that closer to the end of the joist because if the shorter remaining center portion of the beam (hole to end of joist) was too short it could fail in shear and the top and bottom of the 2x10 would act as independent 4" beams (weak).

I'm not a structural or mechanical engineer, who would obviously be more qualified to determine this, but I am an engineer.