Find the length of the shortest path that connects points A and B

Question

Let A be the point (3,5) and let B be the point (6,6). Find the length of the shortest path that connects points A and B if the path must touch the x-axis. The minimal possible length is= ?

Answer 1

The most straightforward way of solving this problem is to cast it as an optimization problem in differential calculus.

 

Since the path must touch the x axis, the path will include a point with coordinates (x,0) for some value of x.   To be the shortest path it will touch only at one point and not go below the x axis.  Thus to be the shortest path, the sum of the distance between (3,5)   and ( x, 0)   and the distance between (x,0) and (6,6) must be a minimum.   The function to be minimized  is thus

 

     D(x) =  sqrt( (3-x)² + 25) +  sqrt((6-x)²+ 36)

 

   As is standard,   we take the derivative of D(x) and set to zero .  This results in

 

    (x-3) /sqrt((3-x)² + 25) = (6-x) /sqrt((6-x)²+ 36)

 

  This equation must be solved for x.   To solve,  cross multiply then square both sides.

    This results in a fourth order equation.  But the terms in x⁴  and x³  cancel out leaving a quadratic equation.

 

   The quadratic equation can be factored yielding the roots   48/11 and -12.   Only the 48/11 makes sense,

   So the point on the x axis is  (48/11, 0)