Regarding whether to add a preposition in infinitive phrase served as attributive

The two sentences I encountered when I read grammar book are as follows:

1. I had no place to live in.



2. A good place to eat is the Sichuan restaurant around the corner.

Regarding the 1st sentence, the grammar book says the infinitive phrase to live in serves as attributive to modify place. The sentence can be changed into: I had no place in which I can live. And there must be a preposition in to form the phrase to live in the place.

Thus, if the rule of the grammar book is right, there should be a preposition at right after eat and the sentence will be A good place to eat at is the Sichuan restaurant around the corner for the 2nd one, which makes up the
phrase A good place at which we can eat.

So, what do I want to know now is which one is right or both of them are correct?