Circle

Equation of family of circles is x²+y² =r²  

For  line y = x+1 to cut the circles in distinct points roots of the equation obtained by solving them simultaneously

so x²+x²+1+2x = r²

so 2x²+2x+1-r²=0

For the equation to have distinct roots D>0

4-8(1-r²)>0

-4+8r²>0

or r²>1/2

or r >.707

Now a+d =1 and a >.707

so d will be in interval (0,1/4) but   b) (0,122) c) (0,224) are supersets of a so check this question