Question

Find the ratio in which the point P(m, 6) divides the join of A(−4, 3) and B(2, 8). Also, find the value of m.

Answer 1

Let the point P(m, 6) divide the line AB in the ratio k : 1.
Then, by the section formula:
$$\begin{gathered} x=\frac{m{x}_2+n{x}_1}{m+n},y=\frac{m{y}_2+n{y}_1}{m+n} \\ \mathrm{The}\mathrm{coordinates}\mathrm{of}P\text{are}(m,6). \\ m=\frac{2k-4}{k+1},6=\frac{8k+3}{k+1} \\ \Rightarrow m(k+1)=2k-4,6k+6=8k+3 \\ \Rightarrow m(k+1)=2k-4,6-3=8k-6k \\ \Rightarrow m(k+1)=2k-4,2k=3 \\ \Rightarrow m(k+1)=2k-4,k=\frac{3}{2} \\ \mathrm{Therefore},\mathrm{the}\mathrm{point}P\mathrm{divides}\mathrm{the}\mathrm{line}AB\mathrm{in}\mathrm{the}\mathrm{ratio}3:2. \\ \mathrm{Now},\mathrm{put}\text{ting}\mathrm{the}\mathrm{value}\mathrm{of}k\mathrm{in}\mathrm{the}\mathrm{equation}m(k+1)=2k-4,\text{we get:} \\ m\left(\frac{3}{2}+1\right)=2\left(\frac{3}{2}\right)-4 \\ \Rightarrow m\left({\displaystyle \frac{3+2}{2}}\right)=3-4 \\ \Rightarrow \frac{5m}{2}=-1\Rightarrow 5m=-2\Rightarrow m=-\frac{2}{5} \\ \mathrm{Therefore},\mathrm{the}\mathrm{value}\mathrm{of}m=-\frac{2}{5} \\ \text{So,}\mathrm{the}\mathrm{coordinates}\mathrm{of}Pare(-\frac{2}{5},6). \end{gathered}$$