If begin{vmatrix} a_1 & b_1 & c_1 a_2 & b_2 | vedclass

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(D) The condition $\begin{vmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{vmatrix} = 0$ is the necessary condition for three lines

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concurrent lines if $\frac{a_i}{a_j} 
eq \frac{b_i}{b_j} 
eq \frac{c_i}{c_j}$ $(i 
eq j)$

Vedclass · Answer

(D) The condition $\begin{vmatrix} a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3 \end{vmatrix} = 0$ is the necessary condition for three lines $a_i x + b_i y + c_i = 0$ $(i = 1, 2, 3)$ to be concurrent.If the lines are not parallel (i.e.,$\frac{a_i}{a_j} 
eq \frac{b_i}{b_j}$ for $i 
eq j$),the vanishing of the determinant implies that the three lines intersect at a single common point.Therefore,if $\frac{a_i}{a_j} 
eq \frac{b_i}{b_j} 
eq \frac{c_i}{c_j}$ for $i 
eq j$,the lines are concurrent.

If $\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 0$,then the lines $a_i x + b_i y + c_i = 0$ $(i = 1, 2, 3)$ represent:

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