Computing the infimum in convex optimization

Question

I am studying optimization from Boyd's book "Convex optimization". In problem 4.25, Linear separation of two sets of ellipsoids, with some hints I concluded that we have the inequalities $\inf_{\|u\|_2 \leq 1} a^TP_i u + a^Tq_i \geq 1$ and $\sup_{\|u\|_2 \leq 1}a^TP_iu + a^Tq_i \leq -1$. Then the solutions say that $\inf_{\|u\|_2 \leq 1} (a^TP_i u + a^Tq_i + b) = -\|a^TP_i\|_2 + a^Tq_i + b$ and $\sup_{\|u\|_2 \leq 1} (a^TP_i u + a^Tq_i + b) = \|a^TP_i\|_2 + a^Tq_i + b$. I am struggling to understand how we got these equalities. Particularly, how did we conclude that $\inf_{\|u\|_2 \leq 1} (a^TP_i u) = -\|a^TP_i\|_2$ and $\sup{\|u\|_2 \leq 1} (a^TP_i u) = \|a^TP_i\|_2$. Can anyone help me understand how this happens?? Thanks!