Question

(i) Find the coordinates of the point that divides the join of A(−1, 7) and B(4, −3) in the ratio 2 : 3.

(ii) Find the coordinates of the point which divides the join of A(−5, 11) and B(4, −7) in the ratio 7 : 2.

Answer 1

(i) The end points of AB are A(−1, 7) and B(4, −3).
Therefore, (x₁ = −1, y₁ = 7) and (x₂ = 4, y₂ = −3)
Also, m = 2 and n = 3
Let the required point be P(x, y).
By section formula, we get:
$$\begin{gathered} x=\frac{\left(m{x}_2+n{x}_1\right)}{\left(m+n\right)},y=\frac{\left(m{y}_2+n{y}_1\right)}{\left(m+n\right)} \\ \Rightarrow x=\frac{\left\{2\times 4+3\times \left(-1\right)\right\}}{2+3},y=\frac{\left\{2\times \left(-3\right)+3\times 7\right\}}{2+3} \\ \Rightarrow x=\frac{8-3}{5},y=\frac{-6+21}{5} \\ \Rightarrow x=\frac{5}{5},y=\frac{15}{5} \\ \mathrm{Therefore},x=1\mathrm{and}y=3 \end{gathered}$$
Hence, the coordinates of the required point are (1, 3).

(ii) The end points of AB are A(−5, 11) and B(4, −7).
Therefore, (x₁ = −5, y₁ = 11) and (x₂ = 4, y₂ = −7).
Also, m = 7 and n = 2
Let the required point be P(x, y).
By section formula, we have:
$$\begin{gathered} x=\frac{\left(m{x}_2+n{x}_1\right)}{\left(m+n\right)},y=\frac{\left(m{y}_2+n{y}_1\right)}{\left(m+n\right)} \\ \Rightarrow x=\frac{\left\{7\times 4+2\times \left(-5\right)\right\}}{7+2},y=\frac{\left\{7\times \left(-7\right)+2\times 11\right\}}{7+2} \\ \Rightarrow x=\frac{28-10}{9},y=\frac{-49+22}{9} \\ \Rightarrow x=\frac{18}{9},y=-\frac{27}{9} \\ \mathrm{Therefore},x=2\mathrm{and}y=-3 \end{gathered}$$
Hence, the required point is P(2, −3).