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Question
Find the equation of a line perpendicular to the line \[\sqrt{3}x - y + 5 = 0\] and at a distance of 3 units from the origin.
Solution
The line perpendicular to \[\sqrt{3}x - y + 5 = 0\] is \[x + \sqrt{3}y + \lambda = 0\]
It is given that the line \[x + \sqrt{3}y + \lambda = 0\] is at a distance of 3 units from the origin.
\[\therefore \left| \frac{\lambda}{\sqrt{1 + 3}} \right| = 3\]
\[ \Rightarrow \lambda = \pm 6\]
Substituting the value of \[\lambda\] we get \[x + \sqrt{3}y \pm 6 = 0\] ,which is equation of the required line.
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