question_answer

Find the coordinates of the points of trisection of the line segment joining \[(4,-1)\]and \[(-\,\,2,-\,\,3)\]

A)  \[\left( 2,-\frac{5}{3} \right)\]     

B)  (1, 3)         

C)  \[(5,-6)\]                       

D)  (3, 7)      

Correct Answer: A

Solution :

Let A \[(4,-1)\]and B \[(-\,\,2,-\,\,3)\]be the line segments and points of trisection of the line segment be P and Q. Then,  \[AP=PQ=BQ=k\]                    (Say) \[\therefore \]      \[PB=PQ+QB=2\,k\] and       \[~AQ=AP+PQ=2\,k\] \[\Rightarrow \]   \[AP:PB=k:2\,k=1:2\] and        \[AQ:QB=2\,k:k=2:1\] Since, P divides AB internally in the ratio 1 : 2. So, the coordinate of P are \[\left( \text{By}\,\,\text{using}\frac{{{m}_{1}}+{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}}\text{and}\frac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}} \right)\] \[=\left( \frac{1\times (-\,\,2)+2\times 4}{1+2},\frac{1\times (-\,\,3)+2\times (-1)}{1+2} \right)\] \[=\left( \frac{-\,\,2+8}{3},\frac{-\,\,3-\,\,2}{3} \right)\] \[=\left( \frac{6}{3},\frac{-\,\,5}{3} \right)=\left( 2,-\frac{5}{3} \right)\] and Q divides AB internally in the ratio 2 : 1.