I just took my car to the tire shop to patch a tire that kept deflating very slowly.
Do I need to get and alignment and balance ?
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If you add a patch to a tire, that is off center weight.
If you do not balance the tire the off center weight might make it shake at certain speeds.
You should definitely get that wheel balanced after it is patched.
Depending on the patch material used on the tire to patch it, it may require a re-balance. I have used plugs probably a hundred times and have never had a re-balance done on them and have had zero issues with balance on the tires. The plugs just don't weigh enough to interfere with it.
If you've had an inside the tire patch, depending on how it was applied, it may/may not need to be re-balanced. If they took the tire off the rim, then yes, it would obviously need to be re-balanced. If, on the other hand, they only detached one bead from the rim and fixed the tire with the other side still intact, they would not need to re-balance it as the tire would still be in the same place as it was prior to the patch job. I've seen it done this way before, but I doubt it happens that often. If there are weights on both sides of the tire, they will most likely remove the weights to aid in the bead displacement process. If this was done, there'd need to be a re-balance done as well.
As far as an alignment goes, you don't need one just because of the work done on the tire. The reason I make this distinction is, if it needed an alignment before the tire work was done, you'd still need one after ;-)
Normally, tire technicians remove the tire from the rim to patch it. When the tire is mounted back onto the rim, it normally can cause the weight to be distributed differently. Balancing the wheel after getting it patched is always a best practice.