If k _ i are possible values of k for which lines k x+2 y+2=0,2 x+ k y+3=0 and 3 x+3 y+ k =0 are concurrent, then k _ i has the value

Step-by-step solution

If three lines a ₁ x+ b ₁ y+ c ₁=0 , a ₂+ b ₂ y+ c ₂=0 and a ₃ x+ b ₃ y+ c ₃=0 are aligned & concurrent, then | array lll a₁ & b₁ & c₁ a₂ & b₂ & c₂ a₃ & b₃ & c₃ array |=0 & | array lll k & 2 & 2 2 & k & 3 3 & 3 & k array |=0 & k^2 (k^2-9 )-2(2 k-9)+2(6-3 k)=0 & k^3-9 k-4 k+18+12-6 k=0 & k^3-19 k+30=0 & (k-2) (k^2 42 k-15 )=0 & (k-2)(k+5)(k-3)=0 & k₁=2, k₂=-5 and k₃=3 & k_i=0 aligned