Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 26I / c / Solution

If $a\ne 0$, the first equation defines the ellipsein the $(x_1,x_2)$-plane, a smooth one-dimensional manifold, while $x_3=a{x}_4$ defines a one-dimensional linear subspace in the $(x_3,x_4)$-plane. Their product is a smooth manifold, parametrized byso $\dim X_a=2$.

For $a=0$, positivity of the first equation forces $x_1=x_2=0$, and the second forces $x_3=0$. Hencea smooth line. Therefore