Answer:
Yes, I can prove that the homogeneous equation of second degree ax² + 2hxy + by² = 0 always represent a pair of straight line passing through origin.
Proof: Let us consider two lines represented by equations y=mx1 and y=mx2 such that they pass through origin (0,0).
Then their general form will be given as follows : Ax+By+C=0 where A , B & C are constants . Now if we equate both these equations then it gives us following relation between m1 & m2 i.e., Amx12 -Bm x22=-C or amx12-bm x22=(a/b)c which is nothing but an equation in standard form with coefficients a , b& c being same for both the lines thus proving our statement. The angle between them would be tan inverse( h / √ab ).