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

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Math Expert
Joined: 02 Sep 2009
Posts: 55226
Two concentric circles have radii 1 and 2. Two points on the outer cir  [#permalink]

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23 Apr 2019, 13:56
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Difficulty:

65% (hard)

Question Stats:

50% (02:36) correct 50% (02:15) wrong based on 16 sessions

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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

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Joined: 19 Oct 2018
Posts: 256
Location: India
Re: Two concentric circles have radii 1 and 2. Two points on the outer cir  [#permalink]

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24 Apr 2019, 01:03
1
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

Bunuel wrote:
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

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57622388_567942300359298_2875292403500056576_n.jpg [ 48.08 KiB | Viewed 263 times ]

Intern
Joined: 10 Jan 2019
Posts: 8
Two concentric circles have radii 1 and 2. Two points on the outer cir  [#permalink]

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24 Apr 2019, 02:22
2
Bunuel wrote:
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

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
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Joined: 21 Jan 2016
Posts: 25
Location: India
GMAT 1: 640 Q34 V44
GPA: 2.4
WE: Manufacturing and Production (Manufacturing)
Re: Two concentric circles have radii 1 and 2. Two points on the outer cir  [#permalink]

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24 Apr 2019, 11:39
Bunuel wrote:
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

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.
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New Doc 2019-04-25 00.00.55.jpg [ 336.89 KiB | Viewed 183 times ]

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Re: Two concentric circles have radii 1 and 2. Two points on the outer cir   [#permalink] 24 Apr 2019, 11:39
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