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Let A B C D be a square with side length 100 . A circle with center C and radius C D is drawn. Another circle of radius r , lying inside A B C D , is drawn to touch this circle externally and such that the circle also touches A B and A D . If r=m+n k , where m, n are integers and k is a prime number, find the value of m+n k .

Correct answer

50

Step-by-step solution

aligned & A Q= 2 r & A C=100 2 & A Q+P Q+C P=100 2 & 2 r+r+100=100 2 & r( 2 +1)=100( 2 -1) & r= 100( 2 -1) 2 +1 & r=100(3-2 2 ) & r=300-200 2 =m+n k & m=300, n=-200, k=2 & m+n k = 300-200 2 =50 aligned

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