I think your question is 住在(X) vs 在(X)住介词. 在：可用作介词，表示事情的时间、地点等[in;on;at]，在 should follow with time, location Just like BingLee said 在住X is wrong. You can say 我住在北京三个月了 and 我在北京住了三个月。These two are all right. But there are still some tiny differnces between them. 1)我住在北京三个月了, means you are still in BeiJing. 2)我在北京住了三个月, maybe yes maybe not depends on the context Because we don't have tense in Chinese, so sometimes we use function word such as 了 to express.For example, 吃饭了about to have dinner ，吃了饭had dinner. Hope it helps.

1.我住在北京3个月了。 I have lived in Beijing for 3 months. 2.我家（住）在北京。 My home is in Beijing.

Generally,we use the former pattern more fluently in chinese,and actually both of them refer to "I live in somewhere" or "my home is in sw"

我家住在北京了。 this is wrong. subject + 住 + somewhere. this subject must be a person. ONLY a person can do the action "live", so ur home or ur house can't "live" somewhere. u should say: 我住在北京了 or 我家搬到北京了 or 我搬到北京了 or 我搬到北京住了---they mean I moved to Beijing and i live there now. with the same reason, u cant say 我家住在北京3个月 u should say 我住在北京3个月了。 u must add 了 at the end because this means i have been living in Beijing for 3 months. u can also say 我住在北京3个月了 or 我家搬到北京3个月了 or 我搬到北京3个月了 or 我搬到北京住了3个月了。---------just like above. 我家在北京。 and 我在北京住。 these two phrases are present.-----they mean I live in Beijing.

Burn the one says 我在住（x). we rarely use 我在住（x). u can say 我在（x）住了（time or times） and 我住在(location)