Depending on the tree species, a 70' tree would have a "root bowl" of at least 70' diameter ................... probably much more. Root barriers are quite commonplace now ................ a trench between the tree and the house ........... filled with concrete and with, possibly, a polystyrene face to accommodate "clay heave" movement (don't worry about it!!!)
Please don't be alarmed, but it would put your mind at rest to have a surveyor check out the site. Just killing the tree might cause even more problems as they extract a huge amount of water while living, but ground conditions might alter after the tree has stopped "drinking". A good surveyor will advise.