Question

Find the perpendicular distance from the origin of the perpendicular from the point (1, 2) upon the straight line \[x - \sqrt{3}y + 4 = 0 .\]

Answer 1

The equation of the line perpendicular to \[x - \sqrt{3}y + 4 = 0\] is \[\sqrt{3}x + y + \lambda = 0\]. 

This line passes through (1, 2).

\[\therefore \sqrt{3} + 2 + \lambda = 0\]

\[ \Rightarrow \lambda = - \sqrt{3} - 2\]

Substituting the value of \[\lambda\],we get

\[\sqrt{3}x + y - \sqrt{3} - 2 = 0\]

Let d be the perpendicular distance from the origin to the line \[\sqrt{3}x + y - \sqrt{3} - 2 = 0\]

\[d = \left| \frac{0 - 0 - \sqrt{3} - 2}{\sqrt{1 + 3}} \right| = \frac{\sqrt{3} + 2}{2}\]

Hence, the required perpendicular distance is \[\frac{\sqrt{3} + 2}{2}\]