Who is the origin and also the center of two concentric circles having radii of the inner and outer circle A and B respectively?

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Two concentric circles have radii and . 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?

Solution

Let the center of the two circles be

. Now pick an arbitrary point

on the boundary of the circle with radius . We want to find the range of possible places for the second point,

, such that

passes through the circle of radius . To do this, first draw the tangents from

to the circle of radius . Let the intersection points of the tangents (when extended) with circle of radius be

and

. Let

be the foot of the altitude from

. Then we have the following diagram.

We want to find

, as the range of desired points

is the set of points on minor arc

. This is because

and

are part of the tangents, which "set the boundaries" for

. Since

and

as shown in the diagram,

is a

triangle with

. Thus,

, and the probability

lies on the minor arc

is thus

Who is the origin and also the center of two concentric circles having radii of the inner and outer circle A and B respectively?

Solution

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