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?

For ÐAQP
≥ 60

(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 = 0^{0.}^{}

^{}

__Case-II:__

The two circles cut each other with each of the centres lying
on the circumference of the

other circle:^{}

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In this case APQ is an

**equilateral triangle**, as the sides PA = QA = PQ = radius of either circle
Hence, ÐAQP = 60

^{0}
For ÐAQP = 0

^{0}, they won’t intersect each other.**For 0**

^{0 }<Ð**AQP < 60**

^{0}, the two points P and Q won’t fall within the intersection of the circles.^{0}, the two points P and Q fall within the intersection of the circles.**
Answer (3)**

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