For me, the use of THE or A here is about specificity and whether it is a singularity. A coast would indicate a general idea, much like saying "I live on a hill". So you have some non specific idea about where I live and that could be fine. "Yeah me too, we live on a hill and we get great views too!" But if I were to say "I live on the hill." (as a first reference, rather a specification like "I live on the second biggest hill in the area") what would you think? Well, that there is ONE hill in the area. It is THE hill. Now a feature about coasts is that there is [generally] only one in the area, so it is always going to be referred to as 'the'. If you live in some less typical geographical feature, I bet local people say, "the north coast" etc. Just for another example. We could say "I live on a lake" because there could be many. Or ONCE WE KNOW in what country a person lives in, then they could say "I live near THE capital." EDIT : using "a coast" Google search throws up... it is also possible to have very swift tidal currents on a coast where the tidal range is But I have to admit using "a coast" in a non-technical way is much less common.

"The coast" would normally be understood as a category (not the mountains, not the city) or as the nearest coast. It's similar to the airport, the bank, the hospital. Grammar in Use has several units on articles. "the bank, the airport, the library, the coast, the beach" are covered.

'the coast' means at the sea/ocean, not inland - it's general; not a specific place. :)

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