Hi
Let me attempt to answer your questions.
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when we say sum of radii , does that mean , sum of distance from center to point A(given 2 and 3 in stmt i and ii respectively).
Yes. The set of all points at a distance of x from another given point O(x,y) forms a circle with centre O and radius x. Therefore, in the given question, the set of all possible values of A at a distance of 2 units from (3,4) forms a circle with centre (3,4) and radius of 2. Similarly, the set of all possible values of A at a distance of 3 units from (0,0) forms a circle with centre (0,0) and radius of 3. The sum of the radii, therefore, is the sum of the distances from the two centres to point A ie; 2+3=5.
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Also lets say if the sum was less than this distance , does that mean circle A can be 2 distinct points?
It is unclear what you mean by "this distance". The sum of the radii is the sum of the distances of A from the given points. Therefore, the sum of these two distances
will always be equal to the sum of the radii.
Alternatively, if you mean the distance between the centres [(3,4) and (0,0)], then the reasoning would be as follows.
If the sum of the radii was
less than the distance between the centres, it would mean that the circles would never intersect. Hence, point A can have no possible co-ordinates satisfying both (i) and (ii). We also know neither (i) nor (ii) alone are sufficient. Hope this helps.
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