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How to determine what the unseen object is if a white dwarf is orbiting it?

Imagine you observe a 1M white dwarf orbiting an unseen object with P = 0.02 years and a a = 0.2AU. What is the unseen object? Why? (your explanation should include a calculation)
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How can astronomers determine the difference between “hydrostatic equilibruim” and “just happens to be spherical”?

This is relevant for the definition of a dwarf planet. I presume the answer will be, well, if we can tell the mass of the body and guess the material. I don't find this very satisfactory because ...