# Shortest distance between Y=2X+4 and point P(-8,9). How to find the shortast distance, please write step by step and graph it.

lfryerda | Certified Educator

Consider the line y=2x+4 and the point P(-8,9).  Call the shortest distance from P to the line to be at the intersection point Q on the line.  Then the line PQ must be perpendicular to the line y=2x+4.  Since the line y=2x+4 has slope 2, then the line PQ has slope `-1/2` .  To find the equation of the line PQ, we use the slope and the point to get:

`y=mx+b`

`9=-1/2(-8)+b`   solve for b

`9=4+b`

`9-4=b`

`b=5`

This means the equation of PQ is `y=-1/2x+5` .  To find Q, we need to find the intersection of the lines `y=-1/2x+5` and `y=2x+4` .  Equating the y-values, we get:

`2x+4=-1/2x+5`   multiply by 2

`4x+8=-x+10`   collect like terms

`4x+x=10-8`

`5x=2`

`x=2/5`

Sub into y=2x+4 to get

`y=2(2/5)+4`

`y={4+20}/5=24/5`

The shortest distance from P to the line y=2x+4 is the distance between `(-8,9)` and `(2/5, 24/5)` .  Using the distance formula, we get:

`d=sqrt{(-8-2/5)^2+(9-24/5)^2}`

`=sqrt{42^2/5^2+21^2/5^2}`

`=sqrt{2205}/5`

`={21sqrt5}/5`

This means that the shortest distance between P and the line y=2x+4 is `{21sqrt5}/5.`

The graph is: