## Thursday, 27 October 2011

### Geometry-8 (CAT-2007)

Two circle with centres P and Q cut each other at two distinct points A and B. the circles have the same radii and neither P nor Q falls with in the intersection of the circles. What is the smallest range that includes all possible values of the angle AQP in degrees?
(1)   Between 0 and 90
(2)   Between 0 and 30
(3)   Between 0 and 60
(4)   Between 0 and 75
(5)   Between 0 and 45
Solution follows here:

Solution:

The two extreme cases for this are:

Case-I:

The two circles touch each other with both the points A and B coinciding:
In this case, the points P,A,Q are collinear and hence ÐAQP = 00.

Case-II:
The two circles cut each other with each of the centres lying on the circumference of the
other circle:

In this case APQ is an equilateral triangle, as the sides PA = QA = PQ = radius of either circle
Hence, ÐAQP = 600
For ÐAQP = 00, they won’t intersect each other.
For 00 <ÐAQP < 600, the two points P and Q won’t fall within the intersection of the circles.
For ÐAQP ≥ 600, the two points P and Q fall within the intersection of the circles.