We fix $x \in \mathcal{C}$ such that $\|Ax\| = 1$. Let $B \in \mathrm{SO}(\mathbb{R}^2)$ be a matrix such that $x = B\binom{1}{0}$.\\
a) Show that for all $r \in ]0,1[$ there exists $x_r \in \mathcal{C}$ such that
$$\left\| AB\begin{pmatrix} r & 0 \\ 0 & \frac{1}{r} \end{pmatrix} x_r \right\| > 1$$
b) Show that if $x_r = \binom{y_r}{z_r}$, then $z_r^2 > \dfrac{r^2}{1+r^2}$.