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# Fill in the blank for the following : If a, b, c are in A.P., then the straight lines ax + by + c = 0 will always pass through ____.

$\begin {array} {1 1} (A)\;(1,-2) & \quad (B)\;(1,2) \\ (C)\;(-1, 2) & \quad (D)\;(-1, -2) \end {array}$

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• If a,b and c are in A.P then $\large\frac{a+c}{2}$$=b \quad \Rightarrow a-2b=-c \quad$ (i.e) $2b-a=c$
Step 1 :
It is given that a, b and c are in A.P
$\therefore c = 2b-a$
Substituting for c in the equation.
$ax+by+c=0$, we get,
$ax+by+2b-a=0$
$\Rightarrow a(x-1)+b(y+2)=0$
Let $\lambda = \large\frac{b}{a}$
$\therefore (x-1)+ \lambda (y+2)=0$
This equation is of the form $L_1+\lambda L_2=0$ which represents a straight line through the intersection of the $L_1=0$ and $L_2=0$.
(i.e) $x-1=0 \quad x=1$
$y+2=0 \quad y=-2$
Hence the straight line always passes through (1, -2).
edited Jul 6, 2014