1969 AMC 12 Question 31

Let $OABC$ be a unit square in the $xy$-plane with $O(0,0),$ $A(1,0),$ $B(1,1),$ and $C(0,1).$ Let $u=x^2-y^2$ and $v=2xy$ define a transformation of the $xy$-plane into the $uv$-plane. The transform (or image) of the square is:

**(A)**

**(B)**

**(C)**

**(D)**

**(E)**