Random variables X and Y are independent standard normal random variables. If U=X^2+Y^2 and V= X/Y, obtain the join pdf of U,V.

Firstly I'm given a hint that this mapping is not one to one.

Suppose (x,y)->(u,v) and (a,b) -> (u,v) . Then a/b=x/y and a^2+b^2=x^2+y^2 so I suppose a counter example is (1,1) and
(-1,-1).

Now I thought I could only use jacobian if the mapping was one to one but seemingly not as I'm given a hint to use it.