The weep hole in the water pump will leak if coolant is leaking past the inner seal at the pump shaft. If this is happening, that means that the bushing on that end of the shaft is also getting coolant leaking through it which is bad news for the bushing. Basically means the pump is on its last legs if indeed that is where the leak is coming from.

And on the blown hose, assuming it did not fail from being too old and brittle, is if the engine compression is overpressuring the cooling system. This can hapen if there is a head gasket leak between a cylinder and cooling passage. A cooling system pressure test can detect this (look for pressure rising with engine RPM) or a chemical test of the coolant can also detect if that is the cause of the problem:
http://www.arrowheadradiator.com/head_gasket_or_combustion_leak_test.htm