Question

Find the ratio at which the $y$-axis divides the line segment joining the points $A(-3,-4)$ and $B(1,-2)$.

Answer 1

Step 1: Defining the problem

Let $\mathrm{m}:\mathrm{n}$ be the ratio at which the $y$-axis divides the line segment joining the points $A(-3,-4)$ and $B(1,-2)$.

Let $P(0,y)$ 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,

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

Therefore, $y$-axis divides the line segment joining the points $A(-3,-4)$ and $B(1,-2)$ in the ratio $3:1$.