To solve this problem we can follow two different approaches:

1)      Using Integral Calculus

2)      Converting the problem in Two body system.

However, First approach is fundamental to solve all center of mass problems, but, here we follow second approach for its simplicity and to save time in problem solving.

Now we consider center of main disc as the origin and disc of radius R removed is lying in third and fourth quadrants.

Let us consider that the mass of Circular disc of radius 2R (say D) = M

After cutting the disc of radius R such that it touches the edge of main disc

Mass of disc of radius R (say D1) removed from main disc

                                   = (M/4R²) × R² = M/4

Therefore, mass of the remaining portion of the disc (say D2) = 3M/4

Now consider discs D, D1 and D2

Coordinates for the Center of mass of disc D = (0,0)

Coordinates for the Center of mass of disc D1 = (0,-R)

Let us assume that Coordinates for the Center of mass of disc D2 = (x,y)

Now, it can be considered that entire mass of discs D1 and D2 are concentrated at their center of mass coordinates, that is the problem is converted in two body which are point objects.

Therefore,

Coordinates of D =

[(M/4) × (coordinates of D1)  + (3M/4) × (coordiantes of D2]/(M/4 + 3M/4) = 0

so,  0 = [(M/4) × (-R)  + (3M/4) × y]/ M

or   y = R/3,

similarly, x = 0 (as it should, well evident from symmetry of the problem)