$\quad\ \begin{cases}T_1^2 + T_2^2 = \left(\sqrt2\pi\right)^2\\\left(\dfrac{3}{2\pi}\right)^2 = \dfrac{1}{T_1^2} +\dfrac{1}{T_2^2}\end{cases}$
$\Leftrightarrow \begin{cases}T_1^2 + T_2^2 = 2\pi^2\\\dfrac{9}{4\pi^2} = \dfrac{T_1^2 + T_2^2}{T_1^2T_2^2}\end{cases}$
$\Leftrightarrow \begin{cases}T_1^2 + T_2^2 = 2\pi^2\\\dfrac{9}{4\pi^2} = \dfrac{2\pi^2}{T_1^2T_2^2}\end{cases}$
$\Leftrightarrow \begin{cases}T_1^2 + T_2^2 = 2\pi^2\\9T_1^2T_2^2= 8\pi^4\end{cases}$
$\Leftrightarrow \begin{cases}T_2^2 = 2\pi^2 - T_1^2\\9T_1^2(2\pi^2 - T_1^2)= 8\pi^4\end{cases}$
$\Leftrightarrow \begin{cases}T_2^2 = 2\pi^2 - T_1^2\\9T_1^4- 18T_1^2+ 8\pi^4=0\end{cases}$
$\Leftrightarrow \begin{cases}T_2^2 = 2\pi^2 - T_1^2\\(4\pi^2 - 3T_1^2)(2\pi^2 - 3T_1^2)=0\end{cases}$
$\Leftrightarrow \begin{cases}T_2^2 = 2\pi^2 - T_1^2\\\left[\begin{array}{l}3T_1^2 = 4\pi^2\\3T_1^2 = 2\pi^2\end{array}\right.\end{cases}$
$\Leftrightarrow \begin{cases}T_2^2 = 2\pi^2 - T_1^2\\\left[\begin{array}{l}T_1^2 = \dfrac43\pi^2\\T_1^2 = \dfrac23\pi^2\end{array}\right.\end{cases}$
$\Leftrightarrow \left[\begin{array}{l}\begin{cases}T_1^2 = \dfrac43\pi^2\\T_2^2 = \dfrac23\pi^2\end{cases}\\\begin{cases}T_1^2 = \dfrac23\pi^2\\T_2^2 = \dfrac43\pi^2\end{cases}\end{array}\right.$
$\Leftrightarrow \left[\begin{array}{l}\begin{cases}T_1 =\pm \dfrac{2\pi\sqrt3}{3}\\T_2 =\pm \dfrac{\pi\sqrt6}{3}\end{cases}\\\begin{cases}T_1= \pm\dfrac{\pi\sqrt6}{3}\\T_2 = \pm\dfrac{2\pi\sqrt3}{3}\end{cases}\end{array}\right.$
