If the lines ax + by + c = 0,bx + cy + a = 0, | vedclass

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(C) The given lines are:$ax + by + c = 0$ $(i)$$bx + cy + a = 0$ $(ii)$$cx + ay + b = 0$ $(iii)$Three lines $a_1x + b_1y + c_1 = 0$,$a_2x + b_2y + c_2

Vedclass · Accepted Answer

$a^3 + b^3 + c^3 - 3abc = 0$

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(C) The given lines are:$ax + by + c = 0$ $(i)$$bx + cy + a = 0$ $(ii)$$cx + ay + b = 0$ $(iii)$Three lines $a_1x + b_1y + c_1 = 0$,$a_2x + b_2y + c_2 = 0$,and $a_3x + b_3y + c_3 = 0$ are concurrent if the determinant of their coefficients is zero:$\begin{vmatrix} a & b & c \ b & c & a \ c & a & b \end{vmatrix} = 0$Expanding the determinant:$a(cb - a^2) - b(b^2 - ac) + c(ab - c^2) = 0$$abc - a^3 - b^3 + abc + abc - c^3 = 0$$3abc - (a^3 + b^3 + c^3) = 0$Therefore,$a^3 + b^3 + c^3 - 3abc = 0$.

If the lines $ax + by + c = 0$,$bx + cy + a = 0$,and $cx + ay + b = 0$ are concurrent,then:

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