 # The ratio in which the line segment joining P (x1, y1) Question:

The ratio in which the line segment joining P (x1y1) and Q (x2, y2) is divided by x-axis is

(a) y1 : y2

(b) −y1 : y2

(c) x1 : x2

(d) −x1 : x2

Solution:

Let $C(x, 0)$ be the point of intersection of $x$-axis with the line segment joining $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right)$ which divides the line segment $P Q$ in the ratio $\lambda: 1$.

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

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

Now we will use section formula as,

$(x, 0)=\left(\frac{\lambda x_{2}+x_{1}}{\lambda+1}, \frac{\lambda y_{2}+y_{1}}{\lambda+1}\right)$

Now equate the y component on both the sides,

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

On further simplification,

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