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 Post subject: GMAT Coordinate Geometry
PostPosted: Thu Jul 12, 2012 2:34 pm 
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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.


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 Post subject: Re: math, t.1, qt.26: coordinate geometry, circles
PostPosted: Thu Jul 12, 2012 2:36 pm 
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Quote:
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.


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 Post subject: Re: math, t.1, qt.26: coordinate geometry, circles
PostPosted: Thu Jul 12, 2012 2:38 pm 
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I follow the explanation, but just curious if part of the question is missing OR it should just simply be understood given this question that the points form a circle? I have been told not to assume pieces of information that aren't told (in this case the concept that the points are related to a circle) so it would seem that the question lacks the necessary information to jump to this conclusion - not sure. Please let me know at your convenience, thanks again.


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 Post subject: Re: math, t.1, qt.26: coordinate geometry, circles
PostPosted: Thu Jul 12, 2012 2:53 pm 
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Quote:
I have been told not to assume pieces of information that aren't told
That is correct.

Quote:
I follow the explanation, but just curious if part of the question is missing OR it should just simply be understood given this question that the points form a circle?
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.


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 Post subject: Re: GMAT Coordinate Geometry (Data Sufficiency)
PostPosted: Tue Oct 16, 2012 12:42 pm 
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Makes sense. Thank you!


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 Post subject: Re: GMAT Coordinate Geometry (Data Sufficiency)
PostPosted: Tue Nov 27, 2012 4:34 am 
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I never thought of a circle in that way. But now I definitely will.


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 Post subject: Re: GMAT Coordinate Geometry (Data Sufficiency)
PostPosted: Sun Jan 06, 2013 2:01 am 
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Can you explain how you would would find the coordinates of the point where the circles touch? It is not clear to me.


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 Post subject: Re: GMAT Coordinate Geometry (Data Sufficiency)
PostPosted: Sun Jan 06, 2013 2:18 pm 
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questioner wrote:
Can you explain how you would would find the coordinates of the point where the circles touch?
Image

In many data sufficiency questions it is necessary to solve the problem all the way till the end in order to see if the solution is exactly one value, multiple values or no such value exists. However in this question we have already showed that the two circles cross in exactly one particular point. Thus the system of equations of the two figures will have exactly one solution.
x² + y² = 3²
(x – 3)² + (y – 4)² = 2²


There is also a fast geometrical way to find the coordinates of the point. Our reasoning led us to the image above (we were able to draw the circles crossing in exactly one point). We can easily find the x-coordinate and y-coordinate of the point B by using the property of similar triangles.

Image

Triangles OBC and OAD are similar. So their corresponding sides are proportional.
OB : OA = OC : OD
3 / (3 + 2) = x / 3
x = 9/5

In the same way, using the similar triangles OBC and OAD, we can find y = 12/5.

Thus B is (9/5, 12/5) .


There are other possible ways to find the coordinates of the point B. But in this problem it is important to see that the two circles define exactly one particular point and thus the solution of the system of the circles's equations will yield exactly one pair of values (x, y) . There is no need in finding the exact values in this question.
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