Question

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

Answer 1

We know that the section formula states that if a point $P(x,y)$ lies on line segment $AB$ joining the points $A(x_1,y_1)$ and $B(x_2,y_2)$and satisfies $AP:PB=m:n$, then we say that $P$ divides internally $AB$ in the ratio $m:n$. The coordinates of the point of division has the coordinates;

$P=(\frac{{mx}_2{+nx}_1}{m+n},\frac{{my}_2{+ny}_1}{m+n})$

Let $P(-2,3)$ divides the line segment $AB$ joining the points $A(-3,5)$ and $B(4,-9)$ in the ratio $m:n$, then using section formula we get,

$P=(\frac{{mx}_2{+nx}_1}{m+n},\frac{{my}_2{+ny}_1}{m+n})$

$\Rightarrow (-2,3)=(\frac{(m\times 4)+(n\times -3)}{m+n},\frac{(m\times -9)+(n\times 5)}{m+n})$

$\Rightarrow (-2,3)=(\frac{4m-3n}{m+n},\frac{-9m+5n}{m+n})\Rightarrow -2=\frac{4m-3n}{m+n}$

$\Rightarrow -2(m+n)=4m-3n$

$\Rightarrow -2m-2n=4m-3n$

$\Rightarrow 4m+2m=3n-2n$

$\Rightarrow 6m=n$

$\Rightarrow \frac{m}{n}=\frac{1}{6}$

$\Rightarrow m:n=1:6$

Hence, the point $(-2,3)$ divides the line segment joining the points $(-3,5)$ and $(4,-9)$ in the ratio $1:6$.