Solve x^{2}+y^2+2ax=4a^2(x^2+y^2)

$$x ^ { 2 } + y ^ { 2 } + 2 a x = 4 a ^ { 2 } ( x ^ { 2 } + y ^ { 2 } )$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{\sqrt{4x^{4}+x^{2}+4y^{4}+8\left(xy\right)^{2}}-x}{4\left(x^{2}+y^{2}\right)}\text{; }a=\frac{\sqrt{4x^{4}+x^{2}+4y^{4}+8\left(xy\right)^{2}}+x}{4\left(x^{2}+y^{2}\right)}\text{, }&x\neq iy\text{ and }x\neq -iy\\a=-\frac{x^{2}+y^{2}}{2x}\text{, }&\left(x=iy\text{ or }x=-iy\right)\text{ and }y\neq 0\\a\in \mathrm{C}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{a^{2}+8\left(ay\right)^{2}-y^{2}-16y^{2}a^{4}}-a}{1-4a^{2}}\text{; }x=-\frac{\sqrt{a^{2}+8\left(ay\right)^{2}-y^{2}-16y^{2}a^{4}}+a}{1-4a^{2}}\text{, }&a\neq \frac{1}{2}\text{ and }a\neq -\frac{1}{2}\\x=\frac{y^{2}\left(4a^{2}-1\right)}{2a}\text{, }&a=\frac{1}{2}\text{ or }a=-\frac{1}{2}\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{\sqrt{4x^{4}+x^{2}+4y^{4}+8\left(xy\right)^{2}}-x}{4\left(x^{2}+y^{2}\right)}\text{; }a=\frac{\sqrt{4x^{4}+x^{2}+4y^{4}+8\left(xy\right)^{2}}+x}{4\left(x^{2}+y^{2}\right)}\text{, }&y\neq 0\text{ or }x\neq 0\\a\in \mathrm{R}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{a^{2}+8\left(ay\right)^{2}-y^{2}-16y^{2}a^{4}}-a}{1-4a^{2}}\text{; }x=-\frac{\sqrt{a^{2}+8\left(ay\right)^{2}-y^{2}-16y^{2}a^{4}}+a}{1-4a^{2}}\text{, }&|a|\neq \frac{1}{2}\text{ and }|y|\leq \frac{|a|}{|4a^{2}-1|}\\x=\frac{y^{2}\left(4a^{2}-1\right)}{2a}\text{, }&|a|=\frac{1}{2}\end{matrix}\right.$$