What are the coordinates of the point A?
(1) A is 2 units away from (3, 4).
(2) A is 3 units away from (0, 0).

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.


(C) All the points that lie at some specific distance from a given one are situated on the circle, which center is the given point and the radius is the distance. Therefore each statement by itself defines a circle (infinitely many points). So each statement by itself is NOT sufficient.

If we use the both statements together, we have two circles. Point A belongs to each one of them. Two circles can cross each other in one or two points. (Two circles can also have no common points at all, but it is NOT our case). The distance between the centers, (0, 0) and (3, 4), is √(3² + 4²) = √25 = 5. It equals to the sum of the radii, 2 + 3 = 5. Therefore the circles touch each other (cross) in exactly ONE point. Clearly, the coordinates of this point can be found by solving the system of the two equations of the circles. However we do NOT need to actually calculate the coordinates.

Statements (1) and (2) taken together are sufficient to answer the question, even though neither statement by itself is sufficient. The correct answer is C.
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It is difficult to understand that the question is talking about circles here or anything related to circles.

What is a circle?
It has its center and all points of the circle are on the fixed distance from that center. The distance is called a radius.

So when a statement tells us "A is 3 units away from (0, 0)" it means that A is some point of the circle, which center is (0, 0) and the radius is 3.

Thus when you see "… the fixed distance from the given point …" you should imagine a circle.

That is correct.

The statement "A is 2 units away from (3, 4)" implies that A is a point on the circle, which center is (3, 4) and its radius is 2 units. So we are not given this fact directly, but deduce it from the statement.

Why is it so? Take any point on the circle. It will be exactly 2 units away from (3, 4). Take any point, which is not on the circle, and its distance to (3, 4) will be more or less than 2 units.


Any time you are given "A is r units away from B" on a plane, then you know that A lies somewhere on the circle, which radius is r units and the center is B. If point A is fixed and point B is unknown, then this phrase means that B lies somewhere on the circle, which radius is r and the center is A.