C is a circle with center D and radius 2. E is a circle with center F

IMO, you can not say statement 1 is sufficient without knowing the exact value of R. Case in point R=3 or R=0.3

You will get 2 different answers for the above 2 values of R.

Here's my POV on it:

No matter what R is, the circles won't touch if the distance between the centers of the two circles is greater than combined radii, R+2, in this case (R being the radius of circle F).

However, statement 1 says that the distance between the centers is R+1, which is shorter than the combined distance of the radii. Therefore, no matter what R is, R+1 is always less than the distance between the two radii and therefore: the circles have to intersect at some point.

Given the two cases: If R=3, the circles are 3+1=4 units apart. The radii are a combined 5 units, therefore the circles intersect.

If R=0.3, the circles are 0.3+1=1.3 units apart. The radii are a combined 2.3 units, therefore the distance is still smaller than the combined length of the radii and thus the circles intersect.

Whatever the value of R, the combined length of the radii will be (R+2)-(R+1)=1 unit larger than the distance between the circles.

For R=0.3, the radii will add up to 2.3 but as the centers will only be 1.3 units apart, the smaller circle will completely lie and not tangentially inside the bigger circle. Hence 0 common points.

Ohhh okay, I get it now, I was wrongly attributing the extra +1 to R as part of the radius. Am I correct in saying that we need to know if R>0.5 to confirm if the circles touch? If R is > or = to 0.5, then the center F is at least 1.5 units away from center D and the opposite end of circle E will touch some point of circle C given that it's at least 1.5+0.5=2 units from center D

Yes, exactly. You need R\(\geq\) 0.5 (and not just R>0.5 as R=0.5 will also give you 1 common points, the point of tangency) is what you need to be absolutely sure. No matter what value of R you assume > 0.5, it will always lead to 2 intersecting circles as the distance between the centers will be 1+R. Additionally, although statement 2 wasnt sufficient on its own, it should provide you the hint that something is going on with this particular value of R=3.

In GMAT DS questions, both statements should lead to the same unique answer. You can not have 2 contradictory unique answers.

Hope this helps.

Awesome, thanks for the explanation!

I will go with option C.

statement 1 : if D = 2R < 1 then the answer will be NO. And if D = 2R >=1 then the answer is Yes. There is no definite answer , and hence not sufficient.
Statement 2 : This is clearly not sufficient. These two circle may intersect or may be independent completely.

combination: from statement one we have to check whether 2R< 1 ? And from statement 2 we have R = 3 i.e 2R = 6 and Is 6 < 1. The answer is no. Hence the circle will intersect. Correct option C.

I guess this solution works for circles external to each other......................

This could be a very silly doubt, but when another circle is completely inside the circle, then automatically don't you get common points on both circles ?