## Least Distance of Line From Origin

Example: Find the least distance between the line

\[3x+2y=6\]

.Rearranging the equation to make

\[y\]

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

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

where \[c\]

is the \[y\]

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,

\[y=\frac{2}{3} \times \frac{18}{13}=\frac{12}{13}\]

.The distance is then, using Pythagoras Theorem,

\[d= \sqrt{(18/13)^2+(12/13)^2}=\frac{6 \sqrt{13}}{13}\]