Question

In what ratio does the point $\left(\frac{24}{11},y\right)$ divide the line segment joining the points P(2, –2) and Q(3, 7)? Also, find the value of y. [CBSE 2017]

Answer 1

Let the point P$\left(\frac{24}{11},y\right)$ divides the line PQ in the ratio k : 1.
Then, by the section formula:
$$\begin{gathered} x=\frac{m{x}_2+n{x}_1}{m+n},y=\frac{m{y}_2+n{y}_1}{m+n} \\ \mathrm{The}\mathrm{coordinates}\mathrm{of}\mathrm{R}\text{are}\left(\frac{24}{11},y\right). \\ \frac{24}{11}=\frac{3k+2}{k+1},y=\frac{7k-2}{k+1} \\ \Rightarrow 24(k+1)=33k+22,y(k+1)=7k-2 \\ \Rightarrow 24k+24=33k+22,yk+y=7k-2 \\ \Rightarrow 2=9k \\ \Rightarrow k=\frac{2}{9} \end{gathered}$$
$$\begin{gathered} \mathrm{Now}\mathrm{consider}\mathrm{the}\mathrm{equation}yk+y=7k-2\mathrm{and}\mathrm{put}k=\frac{2}{9}. \\ \Rightarrow \frac{2}{9}y+y=\frac{14}{9}-2 \\ \Rightarrow \frac{11}{9}y=\frac{-4}{9} \\ \Rightarrow y=\frac{-4}{11} \\ \mathrm{Therefore},\mathrm{the}\mathrm{point}\mathrm{R}\mathrm{divides}\mathrm{the}\mathrm{line}\mathrm{PQ}\mathrm{in}\mathrm{the}\mathrm{ratio}2:9. \\ \text{And,}\mathrm{the}\mathrm{coordinates}\mathrm{of}R\mathrm{are}\left(\frac{24}{11},\frac{-4}{11}\right). \end{gathered}$$