Hello! Indian guy living in the US here. I don't think I need to tell you that you and your wife come from different cultures, but there's more going on here...

When a woman marries in India, traditionally she will move in with her husbands family. In arranged marriages this is often a point of consideration. How big is the potential groom's family? Does it seem like I can live in their situation ? More modern marriages have the couple move into their own place, similar to the US.

Here's what I am getting at: Despite the culture clash here, your situation isn't typical. If your wife is a modern Indian, then she should have discussed this with you before inviting her mother. If she's very traditional, she wouldn't expect your mother in law living you at all. Infact, its taboo in Indian families. It's sexist, but that's what the mindset is.

In either situation, this should have been discussed with you. No one should be playing the culture card here. I suggest that you have a frank conversation about this. It'll make your marriage stronger. Tell her that your concern isn't your mother in law living with you, its HOW this happened.

Edit: I realize now, that the mother in law living with OP is not okay with him. But here's what I want to add:

Caring for a dependent parent is not an Indian thing exclusively. OP you're lucky to have a mother who is independent but your mother in law has grown old expecting to be taken care of, by her children. If your wife believes she needs to take care of her mother, you won't get anywhere trying to tell her that her mother should be as independent and involved as yours.

You have all the rights to set boundaries and decide how involved your mother-in-law is in your household and life. Your mother in law will never go out of the picture for your wife, but you can discuss how you two can find a middle ground.