Use the elimination method to find all solutions of the system
\[
\left\{\begin{array}{l}
3 x^{2}-y^{2}=11 \\
x^{2}+4 y^{2}=8
\end{array}\right.
\]
The four solutions of the system are: $(-a,-b),(-a, b),(a,-b)$, and $(a, b)$ with positive numbers
\[
a=
\]
and $b=$

Step 1 ：Given the system of equations: \[\left\{\begin{array}{l} 3 x^{2}-y^{2}=11 \\ x^{2}+4 y^{2}=8 \end{array}\right.\]

Step 2 ：Add the two equations together to eliminate \(y^2\), resulting in a new equation: \[4x^2 + 3y^2 = 19\]

Step 3 ：Solve the new equation for \(x\), yielding two solutions: \[x = \pm \sqrt{\frac{19 - 3y^2}{4}}\]

Step 4 ：The solutions for \(y\) are \(\pm 1\). Substitute \(y = 1\) into the equation for \(x\) to find the positive solution for \(x\).

Step 5 ：The positive solution for \(x\) is \(2\). Therefore, the four solutions of the system are \((-2,-1),(-2, 1),(2,-1)\), and \((2, 1)\).

Step 6 ：Final Answer: \(a = \boxed{2}\) and \(b = \boxed{1}\)