All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
If your mother is wealthy, there is nothing wrong with asking her to contribute to the costs of the car. However, if she does not have surplus wealth beyond her needs, you should not ask her to do so.
It is incumbent on you to take care of her, obey her in what is permissible, and fulfill her needs to the best of your ability.
Our advice to you is to strive in showing dutifulness to your mother, being kind to her, and endeavoring to make her pleased with you by using your car to fulfil her needs, no matter how much money it would cost you, as long as you can afford it without incurring harm on yourself, and you should do that for the sake of Allah, The Exalted, aspiring to His rewards.
You should rejoice at the blessings of your dutifulness towards your mother which you will see in this worldly life and the Hereafter. Showing dutifulness to one’s mother is one of the dearest deeds to Allah, The Exalted, the fastest in being rewarded, and the most likely to qualify its doer to have his provisions and lifespan blessed. It was narrated on the authority of Ibn ‘Abbaas that he said: “…I do not know of any (better) deed that brings one closer to Allah than dutifulness to the mother.” [Al-Bukhaari in Al-Adab Al-Mufrad]
Allah Knows best.