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

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,

& {{x}_{1}}=5,{{y}_{1}}=5 \\\ & {{x}_{2}}=2,{{y}_{2}}=-1 \\\ & m=2,n=1 \\\ \end{aligned}$$ Now putting these values in the equation $$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{aligned} & 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{aligned}$$ **Thus we have found out the point $$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.