Solve {l}{i^2=c_1^2+c_2^2}{text{Solvefor}atext{where}}{a={(2/3c_{2})}^2}

$$\left. \begin{array} { l } { i ^ {2} = c_{1} ^ {2} + c_{2} ^ {2} }\\ { \text{Solve for } a \text{ where} } \\ { a = {(\frac{2}{3} c_{2})} ^ {2} } \end{array} \right.$$

Solve for c_1, c_2, a

$$\left\{\begin{matrix}\\c_{1}=i\text{, }c_{2}=0\text{, }a=0\text{; }c_{1}=-i\text{, }c_{2}=0\text{, }a=0\text{; }c_{1}=\frac{\sqrt{-9a-4}}{2}\text{, }c_{2}=-\frac{3\sqrt{a}}{2}\text{, }a\neq 0\text{; }c_{1}=-\frac{\sqrt{-9a-4}}{2}\text{, }c_{2}=-\frac{3\sqrt{a}}{2}\text{, }a\neq 0\text{; }c_{1}=\frac{\sqrt{-9a-4}}{2}\text{, }c_{2}=\frac{3\sqrt{a}}{2}\text{, }a\neq 0\text{; }c_{1}=-\frac{\sqrt{-9a-4}}{2}\text{, }c_{2}=\frac{3\sqrt{a}}{2}\text{, }a\neq 0\text{, }&\text{unconditionally}\\c_{1}=\frac{\sqrt{-9a-4}}{2}\text{, }c_{2}=-\frac{3\sqrt{a}}{2}\text{, }a\in \mathrm{C}\text{; }c_{1}=-\frac{\sqrt{-9a-4}}{2}\text{, }c_{2}=-\frac{3\sqrt{a}}{2}\text{, }a\in \mathrm{C}\text{; }c_{1}=\frac{\sqrt{-9a-4}}{2}\text{, }c_{2}=\frac{3\sqrt{a}}{2}\text{, }a\in \mathrm{C}\text{; }c_{1}=-\frac{\sqrt{-9a-4}}{2}\text{, }c_{2}=\frac{3\sqrt{a}}{2}\text{, }a\in \mathrm{C}\text{, }&a\neq 0\text{ and }|\frac{arg(a)}{2}-arg(-\sqrt{a})|<\pi \end{matrix}\right.$$