Question

Find the ratio in which y-axis divides the line segment joining the points A(5, –6) and B(–1, 4) Also, find the coordinates of the point of division.

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(0, y) divides the line segment joining the points A(5, –6) and B(–1, 4) in the ratio k : 1.

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

$$\begin{gathered} \left(0,y\right)=\left(\frac{k\left(-1\right)+1\left({\displaystyle 5}\right)}{k+1},\frac{k\left(4\right)+1\left({\displaystyle -6}\right)}{k+1}\right) \\ \Rightarrow \left(0,y\right)=\left(\frac{-k{\displaystyle +}{\displaystyle 5}}{k+1},\frac{4k-6}{k+1}\right) \\ \Rightarrow 0=\frac{-k+5}{k+1}\mathrm{and}y=\frac{4k-6}{k+1} \\ \Rightarrow 0=\frac{-k+5}{k+1} \\ \Rightarrow -k+5=0 \\ \Rightarrow k=5 \\ \mathrm{Now},y=\frac{4k-6}{k+1} \\ \Rightarrow y=\frac{4\left(5\right)-6}{5+1}\left(\because k=5\right) \\ \Rightarrow y=\frac{20-6}{6} \\ \Rightarrow y=\frac{14}{6} \\ \Rightarrow y=\frac{7}{3} \end{gathered}$$

Hence, the y-axis divides the line segment joining the points A(5, –6) and B(–1, 4) in the ratio 5 : 1.
and the coordinates of the point of division are $\left(0,\frac{7}{3}\right)$.