Question 1011414
Outline of the process:
(1) Distance between the two points
(2) Equation of the line with the points
(3) Equation for the description, one-third the distance from either point
(4) Solve the equation to find the unknown coordinate matching the description


One-third the distance from  (2,1) to (5,8):


Their distance {{{sqrt((5-2)^2+(8-1)^2)=highlight_green(sqrt(58))}}}


{{{m=(8-1)/(5-2)=7/3}}}
{{{y-1=(7/3)(x-2)}}}
{{{highlight_green(y=(7/3)x+1/3)}}},  the general point  (x, (7/3)x+1/3).


Point P is one-third distance from  (2,1) to (5,8), here making (2,1) the choice for reference
to start from.
{{{sqrt((x-2)^2+(7x/3+1/3-1)^2)=(1/3)sqrt(58)}}}
---some algebra steps---
{{{(x-2)^2+(7x/3-1/3)^2=58/9}}}
--work through some more steps--
{{{highlight_green(29x^2-32x-9=0)}}}
-
{{{x=(32-32)/58=0}}}   OR  {{{x=64/58=32/29}}}


y-coordinate which properly fits would be, if checking with a sketched graph, is based on {{{x=32/29}}}.
{{{y=7x/3+1/3}}}
{{{y=7(32/29)/3+1/3}}}
{{{highlight_green(y=253/87)}}}


Point P is the point  {{{highlight(x=32/29)}}}  and  {{{highlight(y=253/87)}}}.


The question asked to find k, in the line containing P, being {{{2x-y+k=0}}}
{{{k=y-2x}}}
{{{k=253/87-2(32/29)}}}
{{{highlight(highlight(highlight_green(k=61/87)))}}}