Question

Given coordinates of points \[P=\left( -1,2 \right)\] , \[Q=\left( 5,5 \right)\] , \[R=\left( 2,-1 \right)\] . “Find the coordinates of \[S\] (\[S\] is on the segment \[QR\] ) if the length of the segment \[QS\] is double the length of segment \[SR\]” ?

Answer 1

Hint: This is one of the very common questions of coordinate geometry. According to the problem, \[S\] is some point on \[QR\] such that \[QS=2SR\] . In other words, we can also say that, \[S\] is a point on the line segment \[QR\] , such that \[S\] divides the line \[QR\] internally in the ratio \[2:1\] . We can find the coordinates of point \[S\] by the formula \[S\equiv \left( \dfrac{m{{x}_{1}}+n{{x}_{2}}}{m+n},\dfrac{m{{y}_{1}}+n{{y}_{2}}}{m+n} \right)\] , where \[m:n\] is the ratio according to which the point divides the line and \[Q=\left( {{x}_{1}},{{y}_{1}} \right)\] , \[R=\left( {{x}_{2}},{{y}_{2}} \right)\] .

Complete step by step answer:

Now, moving off to the solution, let us assume that \[S\] divides the line \[QR\] internally in the ratio \[m:n\] . Let us also assume that the coordinates of \[Q=\left( {{x}_{1}},{{y}_{1}} \right)\] and that of \[R=\left( {{x}_{2}},{{y}_{2}} \right)\] . In such a scenario, the coordinates of the point \[S\] can easily be found out by the given formulae which states that: The abscissa (or x coordinate) is given by, \[\dfrac{m{{x}_{1}}+n{{x}_{2}}}{m+n}\] and the ordinate (or y coordinate) is given by, \[\dfrac{m{{y}_{1}}+n{{y}_{2}}}{m+n}\] . Thus, we can write the final coordinates of point \[S\] as,
\[S\equiv \left( \dfrac{m{{x}_{1}}+n{{x}_{2}}}{m+n},\dfrac{m{{y}_{1}}+n{{y}_{2}}}{m+n} \right)\]. As per the assumptions we have taken, we need to plug in the values into the respective equation of \[S\] .
According to our assumptions, we have defined the following terms,
\[\begin{align}
  & {{x}_{1}}=5,{{y}_{1}}=5 \\
 & {{x}_{2}}=2,{{y}_{2}}=-1 \\
 & m=2,n=1 \\
\end{align}\]
Now putting these values in the equation \[S\equiv \left( \dfrac{m{{x}_{1}}+n{{x}_{2}}}{m+n},\dfrac{m{{y}_{1}}+n{{y}_{2}}}{m+n} \right)\] , we find \[S\] as,
\[S\equiv \left( \dfrac{2\left( 5 \right)+1\left( 2 \right)}{2+1},\dfrac{2\left( 5 \right)+1\left( -1 \right)}{2+1} \right)\]
Evaluating this we get,
\[\begin{align}
  & S\equiv \left( \dfrac{10+2}{3},\dfrac{10-1}{3} \right) \\
 & \Rightarrow S\equiv \left( \dfrac{12}{3},\dfrac{9}{3} \right) \\
 & \Rightarrow S\equiv \left( 4,3 \right) \\
\end{align}\]

Thus we have found out the point \[S\] which divides the line segment \[QR\] internally in the ratio \[2:1\] . Thus the point \[S\equiv \left( 4,3 \right)\] .

Note: The above sum can also be done in another method as we can assume the coordinate of \[S\equiv \left( x,y \right)\] . Now we can find the distance between \[QS\] and \[SR\] using the distance formula and then equate \[QS=2SR\] . Then we can solve it using a simple linear equation method. We must also be careful in understanding whether the point divides the line segment internally or externally and act accordingly.