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Question

Consider a region M which contains all the points (x,y) such that x2+y2100 and sin(x+y)0 where x,yR. If the area of region M is mπ sq units, then the value of m is     


Solution

x2+y2100 represents set of all points (x,y) which lies on or inside the circle x2+y2=100.
Let C denote the disk (x,y) with x2+y2100.
Now, sin(x+y)=0 if and only if x+y=kπ for kZ.
So, disk C has been cut by parallel lines x+y=kπ and in between those lines there are regions containing points (x,y) with either sin(x+y)>0 or sin(x+y)<0.

Since, sin(xy)=sin(x+y), the regions containing points (x,y) with sin(x+y)>0 are symmetric with respect to the origin to the regions containing points (x,y) with sin(x+y)<0.


Thus, from the given figure, the area of region M is half the area of disk C, i.e. 50π sq units.


Alternate Solution: 
x2+y2100 represents set of all points (x,y) which lies on or inside the circle x2+y2=100.
Also, sin(x+y)0
0x+yπor2πx+y3πor4πx+y5πand so on 
x[0,π][2π,3π][4π,5π]

Clearly from figure,
Required Area = Shaded Area
Also, Shaded Area + Unshaded Area = Area of circle
By symmetry,
Shaded Area = Unshaded Area 

 Required Area=12(Area of circle)
                              =12×π(102)
                              =50π sq units 

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