Let the line L: √ 2 x+y=α pass through the point of the intersection P (in the first quadrant) of the circle x^2+y^2=3 and the parabola x^2=2 y. Let…

Let the line $\displaystyle \mathrm{L}: \sqrt{2} x+y=\alpha$ pass through the point of the intersection P (in the first quadrant) of the circle $\displaystyle x^2+y^2=3$ and the parabola $\displaystyle x^2=2 y$. Let the line $\displaystyle L$ touch two circles $\displaystyle C_1$ and $\displaystyle C_2$ of equal radius $\displaystyle 2 \sqrt{3}$. If the centres $\displaystyle Q_1$ and $\displaystyle Q_2$ of the circles $\displaystyle C_1$ and $\displaystyle C_2$ lie on the $\displaystyle y$-axis, then the square of the area of the triangle $\displaystyle P Q_1 Q_2$ is equal to $\displaystyle \_\_\_\_$.