The line segment joining points (−3, −4),
Question:

The line segment joining points (−3, −4), and (1, −2) is divided by y-axis in the ratio

(a) 1 : 3

(b) 2 : 3

(c) 3 : 1

(d) 2 : 3

Solution:

Let $\mathrm{P}(0, y)$ be the point of intersection of $y$-axis with the line segment joining $\mathrm{A}(-3,-4)$ and $\mathrm{B}(1,-2)$ which divides the line segment $\mathrm{AB}$ in the ratio $\lambda: 1$.

Now according to the section formula if point a point $P$ divides a line segment joining $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ in the ratio m:n internally than,

$\mathrm{P}(x, y)=\left(\frac{m_{1}+m x_{2}}{m+n}, \frac{m y_{1}+m y_{2}}{m+n}\right)$

Now we will use section formula as,

$(0, y)=\left(\frac{\lambda-3}{\lambda+1}, \frac{-2 \lambda-4}{\lambda+1}\right)$

Now equate the x component on both the sides,

$\frac{\lambda-3}{\lambda+1}=0$

On further simplification,

$\lambda=3$

So $y$-axis divides $\mathrm{AB}$ in the ratio $\frac{3}{1}$