SOLUTION: What is the distance of the shortest path between the line y=2x + 12 and the point (5,3)?

All you need to do is to find the distance from the point (5,3) to the given straight line  y = 2x + 12.


Write equation of the line in EQUIVALENT form  2x - y + 12 = 0.


There is a remarkable formula which ideally suits for this problem.


    Let the straight line in a coordinate plane is defined in terms of its linear equation 

         a*x + b*y + c = 0,

    where "a", "b" and "c" are real numbers, and let P = (,) be the point in the coordinate plane. 

    Then the distance from the point P to the straight line is equal to

        d = .


Regarding this formula, see the lesson
    The distance from a point to a straight line in a coordinate plane
in this site.


Substitute the coefficients  a= 2, b= -1, c= 12,   = 5,  = 3  into the formula to get the distance under the question


    d =  = .


Answer.  The distance from the point to the line is   =  = 8.497  (rounded).