I am installing a French drain around the back side of my house to help protect the perimeter foundation. I have a 31'X 36' patio butted up against part of the foundation. Should I also install the French drain around the patio?
The drain configuration depends on the direction and degree of slope. Yeah, you might need to work around it, unless the patio is positioned on the upward slope, on the side of the yard opposite the outlet(s). Below is a picture depicting where a french drain could be located with respect to a patio and slope. Also, I am assuming the patio is solid concrete and sheds water away from the house. If not, then you will need to run the drain under it, next to the foundation.
A solid concrete patio might present either a problem or it might save some work. The intention of the drain is to protect the foundation. The patio probably doesn't need protection, and if it sheds water far enough away (30 ft sounds good enough) then it will protect the foundation underneath it from water.
There's a good chance that you will have two drain outlets. Also, if the slope is fairly steep, then you won't need a drain on the lower side. The picture shows a few examples given a gentle slope.