"My student lives in London."
Also, in 在住, 在's function is indicating that the action 住 is/was ongoing.
Ex. 1: 我在住在家里。 I am/was living at home. (Note: The first 在 emphasizes the "ongoing" of the action. Also note that while this sentence is grammatically correct, it's rather unusual - please see ex. 2 for an alternative.)
Ex. 2: 我在住家里。 I am/was living at home. (Note: You don't need 在 to be grammatically correct in this case; it's similar to how there's no preposition in "The ghost is haunting the building." and "The problem haunted me.")
Ex. 3: 我正住在家里。 [Right now/then] I am/was living at home. (Note: 正 has the effect of emphasizing the correctness of the statement.)
Ex. 4: 我正住家里。 [Right now/then] I am/was living at home.
Ex. 5: 我正在住在家里。 [Right now/then] I am/was living at home. (Note: With both the first 在 and the 正, both the "ongoing" of the action and the correctness of the statement are emphasized.)
Ex. 6: 我正在住家里。 [Right now/then] I am/was living at home.
Ex. 7: 我的学生住伦敦。 My student(s) is/are/was/were living in London. (While it's correct, there's no guarantee whether Duolingo accepts it.)
Ex. 8: 我在用电脑。 I am/was using a/the computer(s).
的 is usually used to signify possession, but you can express a strong connection between two subjects (possessor and possession) by omitting to include 的 in the sentence. For example 我爸爸 ("my dad") works just as well as 我的爸爸, exept that in the first sentence it is clearly shown that the speaker is close to his dad and loves him very much. Thats it. Love your dads people.