Question 1064977
Line JK is  {{{y-5=7(x-2)}}}
{{{y=7x-14+5}}}
{{{y=7x-9}}}


Using Distance Formula for "the point that partitions the directed line segment JK in a  3:2 ratio",
{{{sqrt((x-2)^2+(y-5)^2)/sqrt((x-4)^2+(y-19)^2)=3/2}}}


Substitute, simplify, and solve first for x.  Choose the solution which would be between J and K, and then evaluate y.


{{{3sqrt((x-4)^2+(y-19)^2)=2sqrt((x-2)^2+(y-5)^2)}}}

{{{3sqrt((x-4)^2+(7x-9-19)^2)=2sqrt((x-2)^2+(7x-9-5)^2)}}}

{{{9((x-4)^2+(7x-28)^2)=4((x-2)+(7x-14))}}}

{{{9((x-4)^2+7(x-4)^2)=4((x-2)^2+7(x-2)^2)}}}

{{{9*8(x-4)^2=4*8(x-2)^2}}}

{{{72(x^2-8x+16)=32(x^2-128x+128)}}}

.
{{{40x^2-448x+1024=0}}}
.
{{{highlight_green(5x^2-56x+128=0)}}}

(Discriminant, {{{576=24^2}}})

{{{x=(56-24)/10}}}

{{{x=32/10}}}

{{{highlight(x=16/5)}}}


Finding y:
{{{y=7x-9}}}

{{{y=7(16/5)-45/5}}}

{{{y=(112-45)/5}}}

{{{highlight(y=67/5)}}}