Define $L_x$ as the line in the complex plane such that the distance from $x$ to $O$ is the shortest distance from $L_x$ to $O$. How do I find an explicit equation for any arbitrary point on $L_x$ in terms of the complex number $x$? Thanks.
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this simply means that the vector $\vec x$ is perpendicular to the line $L_x$: from here you can get a direction for the line, and since it passes through $x$, you also have one point; that's quite enough. – Nick Pavlov Nov 08 '17 at 16:35
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To cut down on unanswered questions, here we go!
As mentioned in the comments, given a line $\ell$ in the plane and a point $z$ in the plane not lying on the line $\ell,$ the shortest distance from $\ell$ to $z$ is the length of the segment from $z$ to $\ell$ perpendicular to $\ell.$ Turning that around, since the shortest distance from $O$ to $L_x$ is the distance from $O$ to $x,$ there is exactly one $x$ in the plane for which we cannot determine the line $L_x.$ (Can you tell which one?)
As long as we avoid the particular bad $x,$ we know that the point $x$ will yield the line $L_x$ through $x$ perpendicular to the segment from $O$ to $x.$
Cameron Buie
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