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

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Aicharya · Accepted Answer

If three lines $\mathrm{a}_1 x+\mathrm{b}_1 y+\mathrm{c}_1=0$, $\mathrm{a}_2+\mathrm{b}_2 y+\mathrm{c}_2=0$ and $\mathrm{a}_3 x+\mathrm{b}_3 y+\mathrm{c}_3=0$ are $\begin{aligned} & 	ext { concurrent, then }\left|\begin{array}{lll} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{array}ight|=0 \ & \Rightarrow\left|\begin{array}{lll} k & 2 & 2 \ 2 & k & 3 \ 3 & 3 & k \end{array}ight|=0 \ & \Rightarrow k^2\left(k^2-9ight)-2(2 k-9)+2(6-3 k)=0 \ & \Rightarrow k^3-9 k-4 k+18+12-6 k=0 \ & \Rightarrow k^3-19 k+30=0 \ & \Rightarrow(k-2)\left(k^2 42 k-15ight)=0 \ & \Rightarrow(k-2)(k+5)(k-3)=0 \ & \Rightarrow k_1=2, k_2=-5 	ext { and } k_3=3 \ & \Rightarrow \sum k_i=0 \end{aligned}$

If $\mathrm{k}_{\mathrm{i}}$ are possible values of $\mathrm{k}$ for which lines $\mathrm{k} x+2 y+2=0,2…

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