A) Is satisfied if \[\left| d-R \right|\le r\]
B) Is satisfied if \[\left| d-R \right|\le 2r\]
C) Is satisfied if \[\left| d-R \right|\ge r\]
D) Is not satisfied at all
Correct Answer: A
Solution :
[a] Given \[R\ge r>0\] and \[d>0\] \[\Rightarrow 0<\frac{{{d}^{2}}+{{R}^{2}}-{{r}^{2}}}{2dR}\le 1\] \[\Rightarrow 0<(d+R-r)(d+R+r)\le 2dR;\] which is true iff \[({{d}^{2}}+{{R}^{2}}-{{r}^{2}})\le 2dR,\] which is true iff \[{{d}^{2}}+{{R}^{2}}-2dR\le {{r}^{2}}\] \[\Rightarrow {{(d-R)}^{2}}\le {{r}^{2}}\] \[\left| d-R \right|\le r,\] which is also \[-r\le (d-R)\le r\] \[\left| d-R \right|\le r,\] which is also \[-r\le (d-R)\le r\]
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