Frobenius endomorphism on supersingular elliptic curve

Question

Let we have supersingular curve $E(\bar{\mathbb{F}_q})$. Is it true that for every point $P$ $q$-Frobenius endomorphism $\pi_q$ can be write as $A + [q]B$ where $P = A + B$? It is true if $E(\bar{\mathbb{F}_q})[r] = G_1 \times G_2$ where $G_1 =Ker(\pi_q - [1])$ and $G_2 = Ker(\pi_q - [q])$ but I can't find reference on this fact(?). (Of course $(r, q) = 1$ because E is supersingular).