 # The ratio in which the line segment joining points A (a1, b1) Question:

The ratio in which the line segment joining points A (a1b1) and B (a2b2) is divided by y-axis is

(a) −a1 : a2

(b) a1 a2

(c) b1 : b2

(d) −b1 : b2

Solution:

Let $\mathrm{P}(0, y)$ be the point of intersection of $y$-axis with the line segment joining $\mathrm{A}\left(a_{1}, b_{1}\right)$ and $\mathrm{B}\left(a_{2}, b_{2}\right)$ which divides the line segment 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,

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

Now we will use section formula as,

$(0, y)=\left(\frac{\lambda a_{2}+a_{1}}{\lambda+1}, \frac{\lambda b_{2}+b_{1}}{\lambda+1}\right)$

Now equate the x component on both the sides,

$\frac{\lambda a_{2}+a_{1}}{\lambda+1}=0$

On further simplification,

$\lambda=-\frac{a_{1}}{a_{2}}$  