Question 1094460
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Thanks for re-posting in the plain text format.


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There is a remarkable formula for the distance from the point (p,q) to the line ax + by + c = 0 in a coordinate plane.


This formula is


    d = {{{abs(a*p + b*q + c)/sqrt(a^2+b^2)}}}.


    (See the lesson <A HREF=https://www.algebra.com/algebra/homework/Vectors/The-distance-from-a-point-to-a-straight-line-in-a-coordinate-plane.lesson>The distance from a point to a straight line in a coordinate plane</A> in this site).


In our case  p = k,  q = {{{k*sqrt(3)}}}, a= 5, b= 3, c= -12.


So,  d = {{{abs(5*k + 3*k*sqrt(3) - 12)/sqrt(5^2 + 3^2)}}} = {{{abs((5+3*sqrt(3))*k - 12)/sqrt(34)}}},


and your equation for the value of "k" is 


{{{abs((5+3*sqrt(3))*k - 12)/sqrt(34)}}} = 4.


Simplify it

{{{abs((5+3*sqrt(3))*k - 12)}}} = {{{4*sqrt(34)}}}.


There are TWO solutions for k:


1)  k = {{{(12 + 4*sqrt(34))/(5 + s*sqrt(3))}}},   and


2)  k = {{{(12 - 4*sqrt(34))/(5 + s*sqrt(3))}}}.
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Solved.


If you want to know how this formula was derived, look into the referred lesson.