Question

Find the coordinates of the point P which divides line segment QR internally in the ratio m:n in the following examples
(i) Q $\equiv$(-5, 8) R$\equiv$(4, -4) and m:n = 2:1

Answer 1

$$\begin{gathered} \mathrm{Given}:\mathrm{Q}\equiv \left(-5,8\right)\mathrm{and}\mathrm{R}\equiv \left(4,-4\right) \\ \mathrm{Here},x_1=-5,y_1=8,x_2=4\mathrm{and}{y}_2=-4. \\ \mathrm{And}m=2\mathrm{and}n=1. \\ \mathrm{Let}P\equiv \left(x,y\right) \\ \mathrm{By}\mathrm{section}\mathrm{formula}\mathrm{for}\mathrm{internal}\mathrm{division}, \end{gathered}$$

$$\begin{gathered} x=\frac{m{x}_2+n{x}_1}{m+n} \\ \Rightarrow x=\frac{2\times 4+1\times \left(-5\right)}{2+1} \\ \Rightarrow x=\frac{8-5}{3} \\ \therefore x=1 \end{gathered}$$ $$\begin{gathered} y=\frac{m{y}_2+n{y}_1}{m+n} \\ \Rightarrow y=\frac{2\times \left(-4\right)+1\times 8}{2+1} \\ \Rightarrow y=\frac{-8+8}{3} \\ \therefore y=0 \end{gathered}$$

$\therefore \text{C}\mathrm{oordinate}\text{s}\mathrm{of}\mathrm{point}\mathrm{P}\equiv \left(1,0\right)$