Point Q lies in the interior of a particular circle.

Question

Point Q lies in the interior of a particular circle. Is point Q the center of the circle?
 
(1) There exist points A, B, and C, all distinct points on the circle’s circumference, such that the distances QA, QB, and QC are identical.
 
(2) At least two different diameters of the circle contain point Q.

Bunuel · Answer

Point Q lies in the interior of a particular circle. Is point Q the center of the circle? 
 
(1) There exist points A, B, and C, all distinct points on the circle’s circumference, such that the distances QA, QB, and QC are identical. Only the center is equidistant from more than two points on the circumference. Hence Q must be the center. Sufficient.
 
(2) At least two different diameters of the circle contain point Q. Two diameters intersect at the center, hence Q must be the center. Sufficient.

Answer: D.

TheKingInTheNorth · Answer

Hi Bunuel ,

I got the option one .

But for option 2 i am confused . 
(2) At least two different diameters of the circle contain point Q . At least two different diameter means the diameter of different length . Then how come point Q becomes the center of the circle . Please explain . if possible by diagram . Thanks

Bunuel · Answer

The diameter of a circle is the length of the line through the center and touching two points on its edge. The lengths of all possible diameters are the same and different diameters cross each other at the center. https://gmatclub.com/forum/download/file.php?id=27780 So, if two different diameters have one point in common, then it must be the center. Untitled.png

Neeraj91 · Answer

Just wanted to understand - if statement 1 said there exist 2 points instead of 3, this statement would be insufficient right? I am just thinking logically, there can be 2 lines drawn from any point which would be equal ?

acegmat123 · Answer

Yes you are right. It has to be equidistant from at least three points.

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syahasa2 · Answer

My approach to the problem.

(1) If there are 3 points which lie on the circle's circumference then it must be true that Q is the center of the circle.
Lets test a case in which Q is NOT the center of the circle.

If Q is NOT the center of the circle, located ANYWHERE inside the circle, we CAN draw another circle inside this one in which Q is the center as you can see below.
In this case the new circle can only be tangent at ONE point of the bigger circle's circumference -or even not be tangent at all- , making it IMPOSSIBLE for other points to be equidistant from the center of the new circle to the big circle's circumference.

(2) Every pair of diameters of the circle intersect at only one point inside the circle. The center of the circle.

KrishnakumarKA1 · Answer

Prompt analysis
Point Q lies inside the circle.

Super set
The answer could be either YES or NO
Translation
We need other information such that we can answer the question based on the property of the circle

Statement analysis
St 1: 3 non collinear points will be equidistant from only one point. If a circle passes through 3 non collinear points,, then the 4 fourth point which will be equidistant from those 3 points will be the centre of the circle.Hence option b,c and e eliminated.

St 2: two diameters intersect at the centre.if the intersection point is q, therefore q is the centre.

Hence option D

mbaapp1234 · Answer

Statement 1: An infinite number of points can be equidistant from two points on a circle, but only the center of a circle can be equidistance from more than 2 points. So Q must be the center.

Statement 2: Diameters must pass through the center and they only intersect at the center, so Q must be the center of the circle if two diameters converge there.

longhaul123 · Answer

I have doubt in the statement 2, It says that the point Q is at least on the 2 diameters , so point Q can be on the diameter also. So its not that Q is the center of the circle. Kindly correct me . I know my concept is somewhere incorrect

Bunuel · Answer

What do you mean by the highlighted part? Two diameters intersect only at one point, at the centre. If Q is on two diameters then it must be the centre. Please re-read the following posts: https://gmatclub.com/forum/point-q-lies ... l#p1382846 https://gmatclub.com/forum/point-q-lies ... l#p1571862 Hope it helps.

Nunuboy1994 · Answer

Statement 1

This scenario only holds true if each of the lines are equidistant, which can only occur if they branch out from the center of the circle

Statement 2

Bunuel  if two lines are not parallel then they intersect right? And in this scenario the lines must clearly intersect at the center?

D

Bunuel · Answer

________________
Yes, that's correct.

Harsh9676 · Answer

Hi Bunuel,

I chose B, because Q can be equidistant to A, B and C if we draw another circle tangent to the original circle.

Please help

Thanks
Harsh

Bunuel · Answer

We are told that A, B, and C are on the circumference of the circle. In your example, A, B, and C cannot simultaneously be on the circumferences of two different circles, because they would have different curvature.

HWPO · Answer

Bunuel, can you elaborate please? My thought process was the same as Harsh's. As far as we are concerned there is only ONE circle, so what's wrong with Harsh sketch? Let's suppose he drew another circle just to prove a point. We can erase the inner circle he drew and keep the points as they are. Doesn't it prove that statement insufficient?

Thanks.

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Point Q lies in the interior of a particular circle.

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