A) \[0\]
B) \[9\]
C) \[1\]
D) \[-9\]
Correct Answer: D
Solution :
Given equation is \[3{{x}^{2}}+xy-{{y}^{2}}-3x+6y+k=0\] Here \[a=3,b=-1,h=\frac{1}{2},g=-\frac{3}{2},\] \[f=3,c=k\] To represent a pair of lines, \[abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0\] \[\therefore \]\[3(-1)(k)+2\times 3\times \left( -\frac{3}{2} \right)\times \frac{1}{2}\] \[-3{{(3)}^{2}}+1{{\left( \frac{-3}{2} \right)}^{2}}-k{{\left( \frac{1}{2} \right)}^{2}}=0\] \[\Rightarrow \] \[-3k-\frac{9}{2}-27+\frac{9}{4}-\frac{k}{4}=0\] \[\Rightarrow \] \[\frac{-13k}{4}-\frac{117}{4}=0\] \[\Rightarrow \] \[k=-9\]
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