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Question
If the lines x + ay + a = 0, bx + y + b = 0 and cx + cy + 1 = 0 are concurrent, then write the value of 2abc − ab − bc − ca.
Solution
The given lines are
x + ay + a = 0 ... (1)
bx + y + b = 0 ... (2)
cx + cy + 1 = 0 ... (3)
It is given that the lines (1), (2) and (3) are concurrent.
\[\therefore \begin{vmatrix}1 & a & a \\ b & 1 & b \\ c & c & 1\end{vmatrix} = 0\]
\[ \Rightarrow \left( 1 - bc \right) - a\left( b - bc \right) + a\left( bc - c \right) = 0\]
\[ \Rightarrow 1 - bc - ab + abc + abc - ac = 0\]
\[ \Rightarrow 2abc - ab - bc - ca = - 1\]
Hence, the value of 2abc − ab − bc − ca is −1
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