Hello, PJ, and welcome to the Grammar Exchange.

The sample sentences in your post are indeed self-explanatory:

The yen is getting stronger.

A strong yen would not harm the Japanese economy.

As you can see, it is the presence of the adjective "strong" that renders the indefinite article possible. However, the definite article can also be used with adjectives:

-  The strong yen has bolstered the country’s purchasing power overseas (Source).

It seems to me that "a" will be used in those cases where the strength (or some other feature) of the currency is hypothetical or not yet true, while "the" will be used when it is a reality.

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