Question
Write the ratio in which the line segment joining points $(2, 3)$ and $(3, -2)$ is divided by $x$-axis.
✓
Answer
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 $A B$ in the ratio $\lambda: 1$.
Now according to the section formula if point a point $P$ divides 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 $A B$ in the ratio 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, $P ( x , y )=\left(\frac{ mx _1+ mx _2}{m+ n }, \frac{ ny _1+ my _2}{m+ n }\right)$
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,
$\lambda=\frac{3}{2} \text { So } x \text {-axis divides } A B \text { in the ratio } \frac{3}{2} .$
Now according to the section formula if point a point $P$ divides 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 $A B$ in the ratio 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, $P ( x , y )=\left(\frac{ mx _1+ mx _2}{m+ n }, \frac{ ny _1+ my _2}{m+ n }\right)$
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,
$\lambda=\frac{3}{2} \text { So } x \text {-axis divides } A B \text { in the ratio } \frac{3}{2} .$
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