Shortest distance between Y=-3x+2 and point U(7,3).  Calculate the shortest distance between each point and the given line, Please show a full explanation and graph, thank you.  

Expert Answers
lfryerda eNotes educator| Certified Educator

To find the shortest distance between U and the line, we need to have a line from U to the line y=-3x+2 that is perpendicular to y=-3x+2.  If we call the intersection point between these two lines Q, then since y=-3x+2 has slope -3, the line UQ has slope `1/3` .  By using U, we get the equation of UQ:





So the line UQ is `y=1/3x+2/3` .

The intersection between UQ and y=-3x+2 is found by equating the lines:

`1/3x+2/3=-3x+2`   multiply by 3

`x+2=-9x+6`   collect like terms

`x+9x=6-2`  simplify




Sub into y=-3x+2 to get:




So the point Q is `(2/5,4/5)` .  The shortest distance is from U to Q.  This is found through the distance formula:





The shortest distance is `{11sqrt10}/5` .  The graph is: