A) \[\frac{1}{5}\]
B) \[-\frac{1}{5}\]
C) \[-\frac{5}{13}\]
D) \[\frac{5}{13}\]
Correct Answer: D
Solution :
The equation of the circle which passes through the points \[(1,0),\,(0,1)\] and \[(0,0)\] is \[{{x}^{2}}+{{y}^{2}}-x-y=0\] ….(i) Given that, the point \[(2k,\,3k)\] is on the circle and form noncyclic circle. Then, it satisfies the Eq. (i) \[{{(2k)}^{2}}+{{(3k)}^{2}}-(2k)-(3k)=0\] \[\Rightarrow \] \[4{{k}^{2}}+9{{k}^{2}}-5k=0\] \[\Rightarrow \] \[13{{k}^{2}}-5k=0\] \[\Rightarrow \] \[k(13k-5)=0\] \[\Rightarrow \] \[k=0\] or \[k=\frac{5}{13}\] Hence, \[k=\frac{5}{13}\]
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