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.

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.

What is the source on this question as it appears to be beyond the scope of the GMAT