IIT-JEE 2012 · Paper 2, Mathematics question 54

The circles touch. Centres $(0,0)$ and $(3,0)$, radii 2 and 1: the distance 3 equals the sum, so they touch externally at $(2,0)$. Their common tangents are $x=2$ and two direct tangents.

Direct tangents pass through the external centre of similitude, which divides the centres externally in ratio $2:1$: $(6,0)$. A line $y=m(x-6)$ at distance 2 from the origin:
$$\frac{6|m|}{\sqrt{1+m^2}}=2\implies m^2=\frac18.$$
The tangents are $x\pm2\sqrt2y=6$.

(D) $x+2\sqrt2y=6$