Write the ratio in which the line segment joining points

Let $P(x, 0)$ be the point of intersection of $x$-axis with the line segment joining $A(2,3)$ and $B(3,-2)$ which divides the line segment AB 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 $\mathrm{m}$ : $\mathrm{n}$ 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,

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

Now equate the y component on both the sides,

$\frac{3-2 \lambda}{\lambda+1}=0$

On further simplification,

$\lambda=\frac{3}{2}$

So $x$-axis divides $\mathrm{AB}$ in the ratio $\frac{3}{2}$