A) \[\,k\in I\]
B) \[k<0\]
C) \[0<k<1\]
D) For two values of k
Correct Answer: D
Solution :
General equation of circle is, \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] It passes through (0,0), (1, 0) and (0, 1); \ \[c=0\] Now \[2g+1=0\Rightarrow g=-\frac{1}{2}\] and \[2f+1=0\Rightarrow f=\frac{-1}{2}\] Hence equation of circle is \[{{x}^{2}}+{{y}^{2}}-x-y=0\] Point \[(2k,\,\,3k)\]lies on the circle \\[4{{k}^{2}}+9{{k}^{2}}-5k=0\] Þ \[13{{k}^{2}}-5k=0\] Þ \[k=0\] or \[k=\frac{5}{13}\].
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