Question

Find the ratio in which the line segment joining $A(1,-5)$ and $B(-4,5)$ is divided by the $x$-axis. Also, find the coordinates of the point of division.

Answer 1

Step 1: Defining the problem

Let $\mathrm{m}:\mathrm{n}$ be the ratio at which the $x$-axis divides the line segment joining the points $A(1,-5)$ and $B(-4,5)$.

Let $P(x,0)$ be the point of division.

Step 2: Compute the ratio $\mathrm{m}:\mathrm{n}$

Section formula gives us the ratio at which a point divides a line segment. If a point $C$ divides a line segment in the ratio $\mathrm{m}:\mathrm{n}$, then

$C(x,y)=\left(\frac{{\mathrm{mx}}_2+{\mathrm{nx}}_1}{\mathrm{m}+\mathrm{n}},\frac{{\mathrm{my}}_2+{\mathrm{ny}}_1}{\mathrm{m}+\mathrm{n}}\right)$

where, $x_1$ and $x_2$ are the $x$-coordinates and $y_2$ and $y_1$ are the $y$-coordinates of the vertices of the line segment.

Thus,

$$\begin{gathered} y=\frac{m{y}_2+n{y}_1}{m+n} \\ \Rightarrow 0=\frac{\mathrm{m}·5+\mathrm{n}·(-5)}{1+(-3)} \\ \Rightarrow 5\mathrm{m}=5\mathrm{n} \\ \Rightarrow \frac{\mathrm{m}}{\mathrm{n}}=\frac{1}{1} \\ \Rightarrow \mathrm{m}:\mathrm{n}=1:1 \end{gathered}$$

Thus, $x$-axis divides the line in the ratio $1:1$

Step 3: Compute the coordinates of $P$

We already know that the $y$-coordinate of the point is $0$.

We can use section formula again to find the $x$-coordinate,

$$\begin{gathered} x=\frac{{\mathrm{mx}}_2+{\mathrm{nx}}_1}{m+n} \\ \Rightarrow x=\frac{1\times (-4)+1\times 1}{1+1} \\ \Rightarrow x=\frac{-3}{2} \end{gathered}$$

Therefore, the coordinates of point $P$ is $\left(\frac{-3}{2},0\right)$.

Therefore, $x$-axis divides the line segment joining the points $A(1,-5)$ and $B(-4,5)$ in the ratio $1:1$ and the coordinates of the point of division is $\left(\frac{-3}{2},0\right)$.