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The line passing through $(1,2,3)$ and having direction ratios given by $ < 1,2,3>$ cuts the $x$ -axis at a distance $k$ from origin. What is the value of $k ?$
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Let the equation of line which is passing through $(1,2,3)$ and having direction ratios $(1,2,3)$ is
$\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}=a$
$x-1=a$
$\begin{aligned} y &-2=2 a \text { and } z-3=3 a \\ \Rightarrow x &=a+1, y=2 a+2 \text { and } z=3 a+3 \end{aligned}$
At $x$ -axis, $y=0$ and $z=0$ $\Rightarrow 2 a+2=0$ and $3 a+3=0$
$\Rightarrow a=-1$ and $a=-1$
$x=(-1)+1=0$
$\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}=a$
$x-1=a$
$\begin{aligned} y &-2=2 a \text { and } z-3=3 a \\ \Rightarrow x &=a+1, y=2 a+2 \text { and } z=3 a+3 \end{aligned}$
At $x$ -axis, $y=0$ and $z=0$ $\Rightarrow 2 a+2=0$ and $3 a+3=0$
$\Rightarrow a=-1$ and $a=-1$
$x=(-1)+1=0$
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