Dear Vedanshu,

Solution:- Join origin (say, 'O') to the point (say, 'P') where both the circles are touching. This distance (i.e., OP) will be 'R' (since, it is the radius of bigger circle).

Now, let 'A' and 'B' be the point at which smaller circle is touching coordinate axes x and y respectively. And, center point of smaller circle be 'C'.

Then, OA = r, and OB = r. And, OACB is forming a square of side 'r'.

So, OC = √((OA)² + (OB)²) = √2 r

Also, OP = OC + CP = √2 r  + r = (√2 +1) r = R

or, r = R / (√2 +1) [ANS]

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Priyansh Bajaj

center  of smaller circle is at (r,r) ...distance from origen = (r²+r²)^1/2 =2^1/2r

 now the distance  from center of small sphere to point where it touches the big circle is r ...       (from diagram)

so the distance of point of contact from the origin is r + 2^1/2r=(2^1/2+1)r ......1

this total distance is the radius of bigger circle so

     r(2^1/2+1) = R

           r =  R/(2^1/2+1)