Prove that the asymptotes of the hyperbola xy=4x+3y are given by x=3,y=4

The joint equation of asymptotes and hyperbola differs by constant. So let (x-3)(y-4)=k be the joint equation of hyperbola. As these represent joint equation of two lines so (delta, that ahg determinant) must be 0. By making the determinant you will get the desired result.