If it does it when you are applying the brake, it is a sign that you need to have your brakes serviced (sounds like a warped rotor or a failed ABS sensor). Otherwise you should have your alignment checked and tires rotated.
Most probably
I would suspect a bent wheel, but it can also be a tire out of balance, loose lug nuts, bad wheel bearing, or worn steering or suspension parts.
Most likely bad tire or a wheel bent, or out of balance
They stop your car. They are round discs that are kind of attached to the wheels and the pads grip the rotors when you apply the brakes to slow down the car. When a rotor becomes warped due to excessive heat, the steering wheel will shake when you apply the brakes.
Replace your steering dampner. Mine did the same thing.
Yes it is just very hard to turn the wheel when you slow down
The steering wheel shimmering could be warped front rotors, when slowing down. The stalling when slowing down could be a bad fuel filter.
turning the steering wheel uses pressure from the power steering pump and puts a load on the engine causing it to slow the rpm. Should rise again when you stop turning the wheel. completely normal.
If the car is only shaking when you are applying the brakes, the most likely reason is warped rotors. The rotors can become warped when they are overheated and then cooled off too rapidly. This can happen when you have hot rotors and then you run over a puddle and the water splashes on them.
Warped rotor,Tie rods, Wheel bearings (hub).
Your vehicle is.. off does not have electrically assisted steering is broken
Sounds like your brake rotor are warped, a brake shop should be able to turn then (smooth them out) for you if there not too bad. do you feel a little shake at during normal breaking? if so then that could be the problem, but because your were going down hill it take more force to slow down so you feel it more.