Okay, lets first completely understand what is being asked here:

C is a circle with center D and radius 2. E is a circle with center F and radius R. Are there any points that are on both E and C?

In order for points to exist that rest on both circles, we need to know where the circles are relative to each other and we need to know the difference in lengh of their radii. Lets look at (1).

(1) The distance from D to F is R+1

This is not sufficient. Suppose R = 5, then there are in fact points that intersect the circumferences of the circles. However, suppose R = 1/100000. Then, E lives inside C and their circumferences have no points of intersection.

(2) R = 3

This is clearly not sufficient. The circles can intersect or not intersect depending on how far their centers are from each other.

The correct answer is C. (1) tells us where the circles are relative to each other and (2) tells us the length of their radii.

Hey man, How is (1) insufficient?. If the distance between the two circles is R + 1 the least possible distance is when R ->0 i.e. R approaches 0. If we, for argument's sake assume that R is 0, then the distance between the two centers is 1. In this case the inner circle's center is exactly at the midpoint of the radius of the outer circle because the distance between the centers is 1 which is half of the radius 2 of the outer circle. This means that the inner circle touches the outer circle at one single point from inside. If we now increase R then there will always be two points of intersection between the two circles for any values of R. So I think the answer is (A) Correct me if I am wrong please, its 3 am in the morning and I haven't had much sleep

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