The three straight lines ax + by = c,bx + cy | vedclass

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Question

(A) The given three straight lines are $ax + by - c = 0$,$bx + cy - a = 0$,and $cx + ay - b = 0$. For these lines to be concurrent,the determinant of

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$a + b + c = 0$

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(A) The given three straight lines are $ax + by - c = 0$,$bx + cy - a = 0$,and $cx + ay - b = 0$. For these lines to be concurrent,the determinant of their coefficients must be zero: $\begin{vmatrix} a & b & -c \ b & c & -a \ c & a & -b \end{vmatrix} = 0$ Taking $-1$ common from the third column: $-\begin{vmatrix} a & b & c \ b & c & a \ c & a & b \end{vmatrix} = 0$ Applying the operation $C_1 	o C_1 + C_2 + C_3$: $-(a+b+c) \begin{vmatrix} 1 & b & c \ 1 & c & a \ 1 & a & b \end{vmatrix} = 0$ For this to hold,either $(a+b+c) = 0$ or the determinant $\begin{vmatrix} 1 & b & c \ 1 & c & a \ 1 & a & b \end{vmatrix} = 0$. The condition for concurrency is $a + b + c = 0$.

The three straight lines $ax + by = c$,$bx + cy = a$ and $cx + ay = b$ are concurrent,if

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