Find the coordinates of the point R on the line segment joining the points P`(-1,3)` and Q `(2,5)` such that PR `=(3)/(5)` PQ.

Explanation

According to the question ,

Given that , PR `=(3)/(5)PQ`
`rArr(PQ)/(PR)=(5)/(3)`
`rArr(PR+RQ)/(PR)=(5)/(3)`
`rArr1+(RQ)/(PR)=(5)/(3)`
`rArr(PQ)/(PR)=(5)/(3)-1=(2)/(3)`
`:. PQ :PR= 2 : 3`
or `:. PR:RQ = 3 : 2`
Suppose , R (x,y) be the point which divides the line segment joining the points P (-1,3) and Q(2,5) in the ration `3 : 2`.
`:. (x,y) ={(3(2)+2(-1))/(3+2),(3(5)+2(3))/(3+2)}`
[`:'` by internal section formula , `:. (x,y) ={(m_(2)x_(1)+m_(1)x_(2))/(m_(1)+m_(2)),(m_(2)y_(1)+m_(1)y_(2))/(m_(1)+m_(2))}]`
`=((6-2)/(5),(15+6)/(5))=((4)/(5),(21)/(5))`
Hence , the required coordinates of the point R is `((4)/(5),(21)/(5))`.