Gọi \(N\left(x;y\right)\Rightarrow\left\{{}\begin{matrix}\overrightarrow{AN}=\left(x-2;y\right)\\\overrightarrow{BN}=\left(x-1;y-2\right)\end{matrix}\right.\)
\(NA=2NB\Leftrightarrow\sqrt{\left(x-2\right)^2+y^2}=2\sqrt{\left(x-1\right)^2+\left(y-2\right)^2}\)
\(\Leftrightarrow x^2-4x+4+y^2=4\left[x^2-2x+1+y^2-4y+4\right]\)
\(\Leftrightarrow3x^2+3y^2-4x-16y+16=0\)
\(\Leftrightarrow x^2+y^2-\frac{4}{3}x-\frac{16}{3}y+\frac{16}{3}=0\)
\(\Leftrightarrow\left(x-\frac{2}{3}\right)^2+\left(y-\frac{8}{3}\right)^2=\frac{20}{9}\)
\(\Rightarrow a+b+R^2=\frac{2}{3}+\frac{8}{3}+\frac{20}{9}=\frac{50}{9}\)