Question

Find the coordinates of the point which divides the join of $(-1,7)$ and $(4,-3)$ in the ratio$2:3.$

Answer 1

Step 1: Define the problem

Let the point $(-1,7)$ be denoted by $P$ and the point $(4,-3)$ be denoted by $Q$. Thus,

$$\begin{gathered} P(x_1,y_1)=(-1,7) \\ Q(x_2,y_2)=(4,-3) \end{gathered}$$

Let the ratio in which the line is divided be denoted as $m:n$. It is given that the line is divided in the ratio$2:3$, thus,

$$\begin{gathered} m=2 \\ n=3 \end{gathered}$$

Step 2: Apply the section formula

The section formula can be used to calculate the ratio in which a point on a given line divides the line. It is given as,

$\mathit{R}\mathbf{(}\mathit{x}\mathbf{,}\mathbf{}\mathit{y}\mathbf{)}\mathbf{=}\mathbf{(}\frac{\mathbf{m}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{n}{\mathbf{x}}_{\mathbf{1}}}{\mathbf{}\mathbf{m}\mathbf{+}\mathbf{n}}\mathbf{,}\frac{\mathbf{m}{\mathbf{y}}_{\mathbf{2}}\mathbf{+}\mathbf{n}{\mathbf{y}}_{\mathbf{1}}}{\mathbf{}\mathbf{m}\mathbf{+}\mathbf{n}}\mathbf{)}$

Where, $R\left(x,y\right)$ is the coordinates of the point which divides the line in the ratio $m:n$ and $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are the coordinates of the endpoints.

So, by using the formula

$\begin{array}{rcl}R(x,y)& =& \left(\frac{3(-1)+2\left(4\right)}{2+3},\frac{3\left(7\right)+2(-3)}{2+3}\right)\\ & =& \left(\frac{-3+8}{5},\frac{21-6}{5}\right)\\ & =& \left(1,3\right)\end{array}$

Hence, the coordinates of the point which divides the join of $(-1,7)$ and $(4,-3)$ in the ratio$2:3$ is $(1,3)$.