Question

In each of the following examples find the co-ordinates of point A which divides segment PQ in the ratio a : b.
(1) P(–3, 7), Q(1, –4), a : b = 2 : 1
(2) P(–2, –5), Q(4, 3), a : b = 3 : 4
(3) P(2, 6), Q(–4, 1), a : b = 1 : 2

Answer 1

Let the coordinates of point A be (x, y).
(1) P(–3, 7), Q(1, –4), a : b = 2 : 1
Using section formula
$$\begin{gathered} x=\frac{2\times 1+1\times \left(-3\right)}{2+1}=\frac{2-3}{3}=\frac{-1}{3} \\ y=\frac{2\times \left(-4\right)+1\times 7}{2+1}=\frac{-8+7}{3}=\frac{-1}{3} \\ \left(x,y\right)=\left(\frac{-1}{3},\frac{-1}{3}\right) \end{gathered}$$

(2) P(–2, –5), Q(4, 3), a : b = 3 : 4
Using section formula
​$$\begin{gathered} x=\frac{3\times 4+4\times \left(-2\right)}{3+4}=\frac{12-8}{7}=\frac{4}{7} \\ y=\frac{3\times 3+4\times \left(-5\right)}{3+4}=\frac{9-20}{3}=\frac{-11}{7} \\ \left(x,y\right)=\left(\frac{4}{7},\frac{-11}{7}\right) \end{gathered}$$

(3) P(2, 6), Q(–4, 1), a : b = 1 : 2
Using section formula
​$$\begin{gathered} x=\frac{1\times \left(-4\right)+2\times 2}{1+2}=\frac{-4+4}{3}=0 \\ y=\frac{1\times 1+2\times 6}{1+2}=\frac{1+12}{3}=\frac{13}{3} \\ \left(x,y\right)=\left(0,\frac{13}{3}\right) \end{gathered}$$