Two circles 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 within 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

• 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 angle AQP = 0.
  
  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, angle AQP = 60
  For angle AQP = 0, they won’t intersect each other.
  For 0 < angle AQP < 60, the two points P and Q won’t fall within the intersection of the circles.
  
  For angle AQP ≥ 60, the two points P and Q fall within the intersection of the circles.
• 9 years agoHelpfull: Yes(1) No(0)
• ans: (3) between 0 & 60.
  The maximum value of angle AQP is 60 degrees, which happens only when the centers of the two circles will overlap with each other (The radii of both the circles is same. when this happens, AB becomes the common chord and PQ is its perpendicular bisector. the angle AQB becomes 120 degrees. and angle AQP is half of AQB. So minimum range must be less than 60 degrees.
• 11 years agoHelpfull: Yes(0) No(1)