Question

C_1 and C_2 are centres of two identical semi circles with radius 'r' units. The ratio of C_1P_1 to PQ is 1:7 , as shown in the figure. Find the radius of the circle inscribed in the region common to the identical semicircles, given that the circle touches the two semicircles and the line segment P_1P_2.

Answer 1

Answer:

Hi,

the figure may look complicated but can be actually taken as one circle divided into two half..

We can work on one semicircle and then multiply the area of shaded region by 2..

join point B with th center of the diameter say, O..

so OBC is a right angle triangle, as B is the midpoint of arc AC...

what will be the area of shaded region..

=area of sector BC - area of triangle OBC..

sector BC is 1/4 of the circle of radius 2, so area = π *2^2/4= π ..

area of OBC= 1/2* (OB*OC)= 2^2/2=2..

so area of one shaded region= π - 2..

area of total shaded region= 2(π - 2)