Correct use of lie or lay in the following context

Solution 1:

You can convey three different ideas by your choice of lay, lie, or both.

Idea #1: The project focused on technologies in which you were then proficient. You may or may not be proficient in those technologies still, and you may or may not have subsequently extended your proficiency to other areas, but the emphasis here is on your strengths at that time, with no strong indication of what your current strengths are:

I based my final year project around web technologies where my strengths [then] lay.

Idea #2: The project focused on technologies in which you are now proficient. You may or may not have been proficient in those technologies at the time you undertook the project, and you may or may not be proficient in other areas as well, but the implication is that your current strengths are in the technologies you focused on in your project:

I based my final year project around web technologies where my strengths [now] lie.

Idea #3: The project focused on technologies in which you were then proficient and in which you continue to be proficient. You may or may not be proficient in other technologies as well, but the implication is that your current strengths are in the technologies you focused on in your project:

I based my final year project around web technologies where my strengths lay (and continue to lie).

Each of these ideas has its strengths and weaknesses, and it may well be that most people reading whichever wording you choose will have only a vague sense of the implications of that wording.

Solution 2:

I think the issue here is not about understanding the conjugation patterns of the two separate verbs, to lie and to lay, but rather it is working out which verb should always be used in the phrase 'my interests ___ in...' or 'his skills ____ in'. In these phrases the interests or skills are in tangible nouns that we can think of as having 'come to rest' or 'reclining' in an area that they will remain for a while. Therefore the correct verb is 'to lie'

Example:

When I was young my interests lay in sport but not any more; now they lie in politics.

OR

The real skill in poker lies in the ability to read people.