Question

In what ratio does the point P(2, 5) divide the join of A(8, 2) and B(−6, 9)?

Answer 1

Let the point P(2, 5) divide AB in the ratio k : 1.
Then, by section formula, the coordinates of P are
$$\begin{gathered} x=\frac{-6k+8}{k+1},y=\frac{9k+2}{k+1} \\ \text{It is given that}\mathrm{the}\mathrm{coordinates}\mathrm{of}PareP(2,5). \\ \Rightarrow 2=\frac{-6k+8}{k+1},5=\frac{9k+2}{k+1} \\ \Rightarrow 2k+2=-6k+8,5k+5=9k+2 \\ \Rightarrow 2k+6k=8-2,5-2=9k-5k \\ \Rightarrow 8k=6,4k=3 \\ \Rightarrow k=\frac{6}{8},k=\frac{3}{4} \\ \Rightarrow k=\frac{3}{4}\mathrm{in}\mathrm{each}\mathrm{case}. \end{gathered}$$
Therefore, the point P(2, 5) divides AB in the ratio 3 : 4.