Question

Find the co-ordinates of the point B which divides segment PQ externally in the ratio m:n
(i) P $\equiv$ ($-$2, $-$4), Q $\equiv$ ($-$2, $-$1), m:n = 7:1
(ii) P $\equiv$ (0, + 10) Q $\equiv$(3, 7), m:n = 2:5

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{Given}:\mathrm{P}\equiv \left(-2,-4\right)\mathrm{and}\mathrm{Q}\equiv \left(-2,-1\right) \\ \mathrm{Here},x_1=-2,y_1=-4,x_2=-2\mathrm{and}{y}_2=-1. \\ \mathrm{And}m=7\mathrm{and}n=1 \\ \mathrm{Let}\mathrm{B}\equiv \left(x,y\right),\mathrm{which}\mathrm{divides}\mathrm{PQ}\mathrm{externally}. \\ \mathrm{By}\mathrm{section}\mathrm{formula}\mathrm{for}\mathrm{external}\mathrm{division}, \end{gathered}$$

$$\begin{gathered} x=\frac{m{x}_2-n{x}_1}{m-n} \\ \Rightarrow x=\frac{7\times \left(-2\right)-1\times \left(-2\right)}{7-1} \\ \Rightarrow x=\frac{-14+2}{6} \\ \therefore x=-2 \end{gathered}$$ $$\begin{gathered} y=\frac{m{y}_2-n{y}_1}{m-n} \\ \Rightarrow y=\frac{7\times \left(-1\right)-1\times \left(-4\right)}{7-1} \\ \Rightarrow y=\frac{-7+4}{6} \\ \therefore y=\frac{-1}{2} \end{gathered}$$

$\therefore \text{C}\mathrm{oordinate}\text{s}\mathrm{of}\mathrm{point}\mathrm{B}\equiv \left(-2,\frac{-1}{2}\right)$

$$\begin{gathered} \left(\mathrm{ii}\right)\mathrm{Given}:\mathrm{P}\equiv \left(0,10\right)\mathrm{and}\mathrm{Q}\equiv \left(3,7\right) \\ \mathrm{Here},x_1=0,y_1=10,x_2=3\mathrm{and}{y}_2=7 \\ \mathrm{And}m=2\mathrm{and}n=5 \\ \mathrm{Let}\mathrm{B}\equiv \left(x,y\right)\mathrm{which}\mathrm{divides}\mathrm{PQ}\mathrm{externally}. \\ \mathrm{By}\mathrm{section}\mathrm{formula}\mathrm{for}\mathrm{external}\mathrm{division}, \end{gathered}$$

$$\begin{gathered} x=\frac{m{x}_2-n{x}_1}{m-n} \\ \Rightarrow x=\frac{2\times 3-5\times 0}{2-5} \\ \Rightarrow x=\frac{6-0}{-3} \\ \therefore x=-2 \end{gathered}$$ $$\begin{gathered} y=\frac{m{y}_2-n{y}_1}{m-n} \\ \Rightarrow y=\frac{2\times 7-5\times 10}{2-5} \\ \Rightarrow y=\frac{14-50}{-3} \\ \therefore y=12 \end{gathered}$$

$\therefore \text{C}\mathrm{oordinate}\text{s}\mathrm{of}\mathrm{point}\mathrm{B}\equiv \left(-2,12\right)$