Find the shortest distance between the line `y=x-2` and the parabola `y=x^2+3x+2.`

Explanation

Let `P(x_(1),y_(1))` be the point closest to the line y=x-2. Then
`(dy)/(dx)|_((x_(1),y_(1))` = Slope of given line
`2x_(1)+3=1`
`orx_(1)=-1andy_(1)=0`
`y=x^(2)+3x+2`

Hence, point (-1,0) is the closest and its perpendicular distance from the line y=x-2 will be the shortest distance. Therefore,
Shortest distance `=(3)/(sqrt(2))`