Question

In what ratio does the $x-axis$ divide the line segment joining the points $(-4,-6)$ and $(-1,7)$
Also, find the coordinates of the point of division.

Answer 1

Using the section formula, if a point $(x,y)$ divides the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$, then

$(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

Let coordinates of that point be $p(\alpha ,0)$ and it divides AB in the ratio $k:1$

By section formula, y-coordinate of $p$.

$0=\frac{7k-6}{k+1}$

$\Rightarrow k=\frac{6}{7}$

$\Rightarrow \frac{AP}{PB}=\frac{6}{7}$

and x-coordinate,

$\alpha =\frac{-4-k}{k+1}=\frac{-4-\frac{6}{7}}{\frac{6}{7}+1}=\frac{-34}{13}$

$\Rightarrow P(\frac{-34}{13},0)$.