Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics V resit / Question 1 / b / Solution

For $u=1$, the inverse parabolic polar coordinates give $y=v$ and $x=(1-v^2)/2$. For $v=1$, they give $y=u$ and $x=(u^2-1)/2$. Thus the requested curves are the upper halves of the parabolasThe first parabola has vertex $(1/2,0)$ and opens to the left; the second parabola has vertex $(-1/2,0)$ and opens to the right. Both pass through $(0,1)$ and are reflections of one another across the $y$-axis.