Two concentric circles have radii 1 and 2. Two points on the outer cir

Question

Two concentric circles have radii 1 and 2. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?

(A) 1/6
(B) 1/4
(C) \(\frac{2 - \sqrt{2}}{2}\)
(D) 1/3
(E) 1/2

(A) 1/6
(B) 1/4
(C)  \frac{2 - \sqrt{2}}{2} 
(D) 1/3
(E) 1/2

nick1816 · Answer

Let a random point on the outer circle A, draw 2 chords, which are tangent to the inner circle, intersect outer circle at B and C respectively. Chord from A will intersect inner circle only when other point lies on the minor arc BC.
probability that the chord joining the two points intersects the inner circle= angle BOC/360

angle BOC= 2* angle BAC

Now sin(OAC)= 1/2=0.5
Angle OAC=30
angle BAC=2*30=60
angle BOC 2*60=120

Probability= 120/360=1/3

Shubham017 · Answer

First select any arbitrary point on the outer now we can select the second point only from that part of the outer circle such that the two selected points are just able to form a tangent to the inner circle this gives us a region from which we can never intersect the inner circle with two points selected on it,so the required region or circumference is total- that part.
ANS D

prashant212 · Answer

Find the angle subtended by the chord when it is tangent to the inner circle. Find how does the chord divides the circle. Find the probability of choosing a point on outer circle as mentioned in the question.

Two concentric circles have radii 1 and 2. Two points on the outer cir

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Two concentric circles have radii 1 and 2. Two points on the outer cir

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