A) \[{{m}_{1}}{{m}_{2}}{{m}_{3}}{{m}_{4}}=1\]
B) \[{{m}_{1}}{{m}_{2}}{{m}_{3}}{{m}_{4}}=-1\]
C) \[{{m}_{1}}{{m}_{2}}{{m}_{3}}{{m}_{4}}=\frac{1}{2}\]
D) \[\frac{1}{{{m}_{1}}}+\frac{1}{{{m}_{2}}}+\frac{1}{{{m}_{3}}}+\frac{1}{{{m}_{4}}}\]
Correct Answer: A
Solution :
Let the points\[\left( {{m}_{i}},\,\,\frac{1}{{{m}_{i}}} \right),\,\,i=1,\,2,\,3,\,4\]lie on the circle\[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] Then, \[\Rightarrow \]\[m_{1}^{2}+\frac{1}{m_{i}^{2}}+2g{{m}_{i}}+\frac{2f}{{{m}_{i}}}+c=0,\,\,i=1,\,2,\,3,\,4\] \[\Rightarrow \]\[m_{i}^{4}+2gm_{i}^{3}+cm_{i}^{2}+2f{{m}_{i}}+1=0\] \[i=1,\,2,\,3,\,4\] \[\Rightarrow \]\[{{m}_{1}},\,\,{{m}_{2}},\,\,{{m}_{3}}\]and\[{{m}_{4}}\]are the roots of the equation \[{{m}^{4}}+2g{{m}^{3}}+c{{m}^{2}}+2fm+1=0\] \[\Rightarrow \] \[{{m}_{1}}{{m}_{2}}{{m}_{3}}{{m}_{4}}=\frac{1}{1}=1\]
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