Learn English – Tense of “lie”/”lay” in a clause

The verb lay is transitive.

You lay something on the table.

The verb lie is intransitive.

You lie on the table when you are operated upon.

The confusion comes because the past tense of lie is lay:

He lay on the table for two hours before he was operated upon.

Few native speakers get this right. Most people would say, "He laid on the table for two hours." Bill Clinton constantly made this mistake in speeches.

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.