A) \[k=4\]
B) \[k=-1\]
C) \[k=\frac{-4}{5}\]
D) \[k=\frac{-22}{5}\]
Correct Answer: C
Solution :
The given equation \[{{x}^{2}}-4xy-{{y}^{2}}+6x+2y+k-0\] ?.(i) represent the point of straight line, if \[\Delta =0,\] where \[\Delta =abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0\] ?..(ii) Comparing Eq. (i) with the following equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0\] We get, \[a=1,\,h=-2,\,b=-1,\,\,g=3,\,f=1,\,c=k\] From Eq. (ii), \[(1)\,(-1)\,(k)+2(1)\,(3)\,(-2)-(1)\,{{(1)}^{2}}-(-1)\] \[{{(3)}^{2}}-(k)\,{{(-2)}^{2}}=0\] \[\Rightarrow \] \[-k-12-1+9-4k=0\] \[\Rightarrow \] \[-5k=4\,\,\Rightarrow \,k=-4/5\]
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