Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 4

Set $X=x^2$ and $Y=y^2$ in the two conic sections. A real intersection requires $X,Y\ge 0$. Solving the resulting linear equations givesFor $a>0$ the denominator and $X$ are positive, so intersections exist exactly when $a\ge 1$. At $a=1$ they are $(\pm 1,0)$; for $a>1$ both choices of sign for each square root give intersections. Consequently