Lesson Find the point on a given straight line closest to a given point in the plane

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Find the point on a given straight line closest to a given point in the plane

The standard and tradition method for solving such problem is via the geometric considerations.
But, amazingly, using general theory of finding the minimum of a quadratic function provides MUCH SHORTER way to the answer.

Problem 1

Find the point on the line y = 4x that is closest to the point P = (1,2).

Solution

On the line (x,4x) we want to find the point closest to (1,2).


Write the distance formula (the square of distance formula)


    d^2 = (x-1)^2 + (4x-2)^2 = 

        = x^2 - 2x + 1 + 16x^2 - 16x + 4 = 

        = 17x^2 - 18x + 5.


The minimum of this quadratic form is at  x =  =  = .


ANSWER.  The closest point is  ( , ) = ( , ).

Problem 2

Find the point on the line y = 5x+1 that is closest to the point (3,5).

Solution

On the line (x,5x+1) we want to find the point closest to (3,5).


Write the distance formula (the square of distance formula)


    d^2 = (x-3)^2 + (5x+1-5)^2 = 

          (x-3)^2 + (5x-4)^2 = x^2 - 6x + 9 + 25x^2 - 40x + 16 = 

                             = 26x^2 - 46x + 25.


The minimum of this quadratic form is at  x =  =  = .


ANSWER.  The closest point is  ( , ) = ( , ).

My other lessons in this site on finding the maximum/minimum of a quadratic function are
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola
    - A rectangle with a given perimeter which has the maximal area is a square
    - A farmer planning to fence a rectangular garden to enclose the maximal area
    - A farmer planning to fence a rectangular area along the river to enclose the maximal area
    - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area
    - Finding the maximum area of the window of a special form
    - Using quadratic functions to solve problems on maximizing revenue/profit
    - When is the best time to sell a pig ?
    - Minimal distance between sailing ships in a sea
    - Advanced lesson on finding minima of (x+1)(x+2)(x+3)(x+4)
    - OVERVIEW of lessons on finding the maximum/minimum of a quadratic function

Use this file/link ALGEBRA-I - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-I.

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