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# State whether the following statement is true or false and justify : The lines $ax + 2y + 1 = 0, bx + 3y + 1 = 0$ and $cx + 4y + 1 = 0$ are concurrent if a, b, c are in G.P

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Toolbox:
• If three lines $a_1x+b_1y+c_1=0 ; a_2x+b_2y+c_2 = 0 ; a_3x+b_3y+c_3 = 0$ are coplanar then $\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} = 0$
Step 1 :
The given lines are:
$ax+2y+1=0 ; bx+3y+1=0 ; cx+4y+1=0$
It is said that the three lines are coplanar.
$\therefore \begin{vmatrix} a & b & c \\ 2 & 3 & 4 \\ 1 & 1 & 1 \end{vmatrix}=0$
On expanding we get,
$a(3-4)-b(2-4)+c(2-3)=0$
$\Rightarrow -a+2b-c=0$
$\Rightarrow 2b=a+c$
This implies that a,b,c are in A.P.
Hence the statement is false.
answered Jul 9, 2014