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A light source at the point (0,16) in the coordinate plane casts light in all directions. A disc (a circle along with its interior) of radius 2 with center at (6,10) casts a shadow on the X axis. The length of the shadow can be written in the form m n where m, n are positive integers and n is square-free. Find m+n .

Correct answer

21

Step-by-step solution

C (x-6)^2+(y-10)^2=2^2 Let tangent be m x-y+16=0 aligned & | 6 m+6 1+m^2 |=2 & 9 m^2+9+18 m=m^2+1 & 8 m^2+18 m+8=0 & 4 m^2+9 m+4=0 & m= -9 17 8 & |x₁-x₂ |= | 16 8 -9+ 17 - 16 8 -9- 17 | & = 128|-9- 17 +9- 17 | (81-17) & =2|2 17 |=4 17 & m+n=21 aligned

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