Question

Find the ratio in which the line segment joining the points $A\left(3,-3\right)$ and $B\left(-2,7\right)$ is divided by $x-axis$. Also find the coordinates of the point of division.

Answer 1

Step1: Finding the Ratio in which the line segment is divided:

Let the points $A\left(3,-3\right)$ and $B\left(-2,7\right)$ is divided by the point $P\left(x,y\right)$ which lies on the $x-axis$, in the ratio $m:n$.

Using Section formula between points $A$ and $B$ $=\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right)=\frac{-2m+3n}{m+n},\frac{7m+(-3)n}{m+n}$

Thus, $P\left(x,y\right)$$=\frac{-2m+3n}{m+n},\frac{7m+(-3)n}{m+n}$…………..$\left(i\right)$

Since the point $P$ lies on the $x-axis$, its $y$ coordinate will be equal to zero.

i.e, $\frac{7m+\left(-3\right)n}{m+n}=0$

$\Rightarrow 7m-3n=0$

$\Rightarrow 7m=3n$

Hence, the ratio in which line segment $AB$is divided by $x-axis$ is $3:7$

Step 2: Finding the coordinates of the point of division:

The coordinates of the point $P$ which lies on $x-axis$ is

Putting values in $\left(i\right)$, we get:

$P(x,y)=P\left(\frac{-2\times 3+3\times 7}{3+7},0\right)=P\left(\frac{15}{10},0\right)$

Therefore, the required ratio and point is $3:7$ and $P\left(\frac{3}{2},0\right)$