Q: the car chakes in reverse or 1st gear only.

There are a couple of possible issues. I am going to assume that the car shakes when you are accelerating in first or reverse, and not idling. If this is the case, it sounds like a driveline vibration.

First and reverse deliver the most torque to the driveline and therefore will magnify any problems. Check your engine and trans mounts. A worn or broken mount can shift the engine/trans out of alignment and cause a vibration. This could also be what you are feeling, sitting still and shifting to first or reverse.

If they are ok, you may have a bad axle, sometimes called a half shaft. The problem most likely lies in the inner joint, (closest to the transmission) of the axle. There are 3 sets of needle bearings. All it takes is some of the bearings to be broken or seized, and the axle will not rotate on center, causing a vibration. Look for torn grease boots for a clue. If you see nothing obvious, the only way to know for sure is to remove the axle, take it apart, and inspect it.

You could also just replace both if you can find no other problems. If you’d like to get this serviced or would like a second opinion on the matter, consider enlisting a qualified technician, like one from YourMechanic. They will be able to diagnose the vibrations firsthand and help you fix it accordingly. Good luck.