Question

In what ratio does the point $P(-2,3)$ divide the line segment joining the points $A(-3,5)$ and $B(4,-9)$ internally?

Answer 1

Given points are $A(-3,5)$ and $B(4,-9)$
Let $P(-2,3)$ divide $AB$ internally in the ratio $l:m$
By the section formula,
$(\frac{{lx}_2+{mx}_1}{l+m},\frac{{ly}_2+l{y}_1}{l+m})=P(-2,3)......\left(1\right)$
Here $x_1=-3,y_1=5,x_2=4,y_2=-9$
$\left(1\right)$ $\Rightarrow$ $(\frac{l\left(8\right)+m\left(3\right)}{l+m},\frac{l(-9)+m\left(5\right)}{l+m})=(-2,3)$
Equating the x-coordinates, we get
$\frac{4l-3m}{l+m}=-2$
$\Rightarrow$ $6l=m$
$\frac{l}{m}=\frac{1}{6}$
i.e., $l:m=1:6$
Hence $P$ divides $AB$ internally in the ratio $1:6$