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co-ordinate geometry

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co-ordinate geometry

by mitaliisrani » Wed Jun 02, 2010 8:38 pm
A and B are two points on a coordinate plane. All the points , in the same plane as A and B , whose distance from B is twice that from A lie on

A) straight line intersecting AB at a point O such that 2AO = BO.
B) a circle with center at a point O on AB such that AO = 2BO.
C) a circle with center at a point O on AB extended such that 4AO = BO.
D) None of these

Please could somebody solve this

OA:C
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by liferocks » Wed Jun 02, 2010 9:04 pm
this has asked the concept of Apollonius circle
You can see the section for Apollonius circle in Wikipedia for the detail explanation
https://en.wikipedia.org/wiki/Circle

Hope this will help.
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by 4GMAT_Mumbai » Wed Jun 02, 2010 9:13 pm
Hi,

If A(x1, y1) and B(x2, y2) are the two points, then the locus of the point in question is:

2 times sqrt [(x - x1) ^ 2 + (y - y1) ^ 2] = sqrt [(x - x2) ^ 2 + (y - y2) ^ 2]

Squaring on both sides,

4 times [(x - x1) ^ 2 + (y - y1) ^ 2] = [(x - x2) ^ 2 + (y - y2) ^ 2]

It is obvious that it is a circle ... In the two minutes that GMAT gives, one can deduce that A is out of the window. B is also out of the window because the ratio of distances (in this choice) is with respect to the center of the circle. But, I am interested with the points on the circumference of the circle.

With 4 or 5 minutes, one might be able to figure out that the answer is C. But, that is only a short term objective. long term objective is to maximise the score in all the 37 questions ... Wondering if there is a shortcut to decide b/w C and D ...

Hope this helps. Thanks.
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by tpr-becky » Thu Jun 03, 2010 12:04 am
What is the source on this question as it appears to be beyond the scope of the GMAT
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