Question 1160833
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            I will present here another way to solve the problem.



<pre>
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 = {{{-b/(2a)}}} = {{{18/(2*17)}}} = {{{9/17}}}.


<U>ANSWER</U>.  The closest point is  ( {{{9/17}}},{{{4*(9/17)}}} ) = ( {{{9/17}}},{{{36/17}}} ).
</pre>

Solved.