I want to put a hole through the joist in the narrowed position in order to run a 50mm shower waste through to the stack

its a 63X200 joist and from what i can measure under the floor broads is at least a 3m span.

 

Its close to the end of the joist which is my concern.

Was wondering if it might be doable if i reinforce the joist either side with thick ply, glued and screwed in place before drilling through the joist and ply

 

There are standard metal fixings from Simpson Strong tie for this have a look here: https://www.strongtie.co.uk/products/detail/i-joist-hole-support/317 for I-joist there must be an equivalent for plain joists - found this one stateside: https://joistrepair.com/pages/applications

Not a problem drilling 50-60mm hole in a 200mm joist as long as it is mid depth. A bit of reinforcement won’t do any harm.