The y-axis is the directrix of the ellipse with eccentricity $e = \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ and the corresponding focus is at $(3, 0)$, equation to its auxiliary circle is

$$\begin{gathered} Directrix : x=0 \\ e=1/2 \\ Focus=(3, 0) ; u\sin g focal directrics property \\ \therefore \sqrt{{(x-3)}^2+y^2}=\frac{1}{2}.\left|x\right| \\ \therefore {(x-3)}^2+y^2=\frac{1}{4}.\left|x\right| \\ \Rightarrow {4(x-3)}^2+4y^2 = 3x^2-24x+4y^2+36=0 \\ \Rightarrow {3(x-4)}^2+4y^2 = 12 \\ \frac{{(x-4)}^2}{4}+\frac{y^2}{3}=1 \dots \left(1\right) \\ \therefore a=2 ; b=\sqrt{3} ; centre (4,0) \\ \Rightarrow auxillary circle is {(x-4)}^2+y^2=4 \end{gathered}$$