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If $a, b$ and $c$ are in $\mathrm{AP}$, then the straight line $a x+2 b y+c=0$ will always pass through a fixed point whose coordinates are
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Verified Answer
The correct answer is:
(1,-1)
Given that, $a$, $b$ and $c$ are in $\mathrm{AP}$
$\therefore \quad 2 b=a+c$
$\Rightarrow \quad c=2 b-a$
and equation of straight line is
$a x+2 b y+c=0$
$\Rightarrow \quad a x+2 b y+(2 b-a)=0$
$\Rightarrow \quad a(x-1)+b(2 y+2)=a \cdot 0+b \cdot 0$
On comparing, we get
$x-1=0 \Rightarrow x=1$
and $\quad 2 y+2=0$
$\Rightarrow \quad y=-1$
Hence, the required fixed point is (1,-1)
$$
$\therefore \quad 2 b=a+c$
$\Rightarrow \quad c=2 b-a$
and equation of straight line is
$a x+2 b y+c=0$
$\Rightarrow \quad a x+2 b y+(2 b-a)=0$
$\Rightarrow \quad a(x-1)+b(2 y+2)=a \cdot 0+b \cdot 0$
On comparing, we get
$x-1=0 \Rightarrow x=1$
and $\quad 2 y+2=0$
$\Rightarrow \quad y=-1$
Hence, the required fixed point is (1,-1)
$$
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