Yes, the ratio can be used to corroborate the double-stranded nature of DNA, however, I doubt that you could use the purine:pyrimidine ratio to determine whether the dsDNA is linear or circular as both forms consist of dsDNA. There are alternative means of testing this, such as actual visulaisation of the DNA using atomic force microscopy (AFM) or electron microscopy (EM). If you knew the sequence, and this sequence was the same for both forms of DNA, then a digest with a single-cutter restriction enzyme would result in two fragments in the linear, and only a single linear fragment of the circular. The two different "topologies" of DNA will also migrate differently within an electrophoresis gel (agarose or acrylamide) for any given size.

You could use the ratio value to determine whether you had a solution of dsDNA and a solution of RNA - but you wouldn't want to as there are far easier ways to do this, not least because RNA contains uracil.

Hope that helps

Jim (MAD Scientist)