1. Tardigrade
  2. Question
  3. Mathematics
  4. Statement-1: If three lines L1: a1x + b1y + c1 = 0, L2: a2x + b2y + c2 = 0 and L3: a3x + b3y + c3 = 0 are concurrent lines, then |a1&b1&c1 a2&b2&c2 a3&b2&b3| = 0. Statement-2: If |a1&b1&c1 a2&b2&c2 a3&b2&b3| = 0 then the lines L1, L2, L3 must be concurrent.

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Q. Statement-1 : If three lines $L_{1} : a_{1}x + b_{1}y + c_{1} = 0, L_{2} : a_{2}x + b_{2}y + c_{2} = 0$ and $L_{3} : a_{3}x + b_{3}y + c_{3} = 0$ are concurrent
lines, then $\begin{vmatrix}a_{1}&b_{1}&c_{1}\\ a_{2}&b_{2}&c_{2}\\ a_{3}&b_{2}&b_{3}\end{vmatrix} = 0.$
Statement-2 : If $\begin{vmatrix}a_{1}&b_{1}&c_{1}\\ a_{2}&b_{2}&c_{2}\\ a_{3}&b_{2}&b_{3}\end{vmatrix} = 0$ then the lines $L_{1}, L_{2}, L_{3}$ must be concurrent.

Determinants

Solution:

If $\Delta = 0$ then two of rows or column are proportional which is possible even if three lines are parallel or two of them are coincident.