Question 1008389
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find the coordinates of the point which divides the line segment from (-1,4) to (2,-3) in to ratio of 3 to 4 
(two solutions)
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        Calculations and the answer in the post by @mananth are totally and fatally incorrect.

        His formulas are conceptually and methodically incorrect; 

        his calculations are partly incorrect and partly incomplete.


        Below is my correct solution.



<pre>
The points are  A = (x1,y1) = (-1,4) and B = (x2,y2) = (2,-3).


Ratio of division is  m:n = 3:4.



If to interpret the ratio 3:4 as starting from A to B,
then the coordinates of the point which divides the given segment are


    x = {{{(m*x2+n*x1)/(m+n)}}} = {{{(3*2+4*(-1))/(3+4)}}} = {{{2/7}}},

    y = {{{(m*y2+n*y1)/(m+n)}}} = {{{(3*(-3)+4*4)/(3+4)}}} = {{{7/7}}} = 1,



If to interpret the ratio 3:4 as starting from B to A,
then the coordinates of the point which divides the given segment are

    x = {{{(m*x1 + n*x2)/(m+n)}}} = {{{(3*(-1)+4*2)/(3+4)}}} = {{{5/7}}},

    y = {{{(m*y1 + n*y2)/(m+n)}}} = {{{(3*4   +4*(-3))/(3+4)}}} = {{{0}}}.


<U>ANSWER</U>.  If to count the ratio 3:4 from A to B, then the division point is  ({{{2/7}}},{{{1}}}).

         If to count the ratio 3:4 from B to A, then the division point is  ({{{5/7}}},{{{0}}}).
</pre>

Solved correctly.