Question

The x-axis divides the line segment joining the points (–4, –6) and (–1, 7) in the ratio _________.

Answer 1

Section formula: if the point (x, y) divides the line segment joining the points (x₁, y₁) and (x₂, y₂) internally in the ratio k : 1, then the coordinates (x, y) = $\left(\frac{k{x}_2+x_1}{k+1},\frac{k{y}_2+y_1}{k+1}\right)$

Let the point P(x, 0) divides the line segment joining the points (–4, –6) and (–1, 7) in the ratio k : 1.

Therefore, using section formula, the coordinates of P are:

$$\begin{gathered} \left(x,0\right)=\left(\frac{k\left(-1\right)+1\left({\displaystyle -4}\right)}{k+1},\frac{k\left(7\right)+1\left({\displaystyle -6}\right)}{k+1}\right) \\ \Rightarrow \left(x,0\right)=\left(\frac{-k{\displaystyle -}{\displaystyle 4}}{k+1},\frac{7k-6}{k+1}\right) \\ \Rightarrow x=\frac{-k{\displaystyle -}{\displaystyle 4}}{k+1}\mathrm{and}0=\frac{7k-6}{k+1} \\ \Rightarrow 0=\frac{7k-6}{k+1} \\ \Rightarrow 7k-6=0 \\ \Rightarrow 7k=6 \\ \Rightarrow k=\frac{6}{7} \end{gathered}$$

Hence, the x-axis divides the line segment joining the points (–4, –6) and (–1, 7) in the ratio 6 : 7.