\Delta^2 = AP^2 = (\overrightarrow{AO}+\overrightarrow{OP}) = a^2 + r^2 - 2 \overrightarrow{OA} . \overrightarrow{OP}

= a^2 - 2 a r \left|\begin{array}{c} \cos \alpha\\ \sin \alpha\\ 0\\ \end{array} . \left|\begin{array}{c} \cos \varphi \cos \lambda\\ \cos \varphi \sin \lambda\\ \sin \varphi\\ \end{array} + r^2

= a^2 - 2 a r \cos \varphi \cos (\alpha -\lambda) + r^2