Least Distance of Line From Origin

The line of least distance between a given line and the origin is perpendicular to that line. Since the line of least distance passes through the origin, we can find the equation of this line, then by solving simultaneous equations, the point of intersection of the line of least distance with the given line. Finally, use Pythagoras Theorem to find the least distance.
Example: Find the least distance between the line  

least distance from origin

Rearranging the equation to make  
  the subject gives  
\[y=- \frac{3}{2} x +3\]
This line has gradient  
\[- \frac{3}{2}\]
  so the perpendicular line has gradient  
\[- \frac{1}{- (3/2)}=\frac{2}{3}\]
The equation of the perpendicular line is  
\[y= \frac{2}{3}x+c\]
  is the  
  intercept - 0 since the line passes through the origin.
The point of intersection of the two lines is the solution to the simultaneous equations
\[y=- \frac{3}{2}x +3\]

\[y= \frac{2}{3}x\]

Subtracting the second from the first gives  
\[0=- \frac{13}{6}+3 \rightarrow x=\frac{3}{13/6}=\frac{18}{13}\]
Then from the second of these equations,  
\[y=\frac{2}{3} \times \frac{18}{13}=\frac{12}{13}\]

least distance from origin

The distance is then, using Pythagoras Theorem,  
\[d= \sqrt{(18/13)^2+(12/13)^2}=\frac{6 \sqrt{13}}{13}\]

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