Question

In what ratio does the point $(-4,6)$ divide the line segment joining the points $A(-6,10)$ and $B(3,-8)$

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 the required ratio be $m:n$

$A(-6,10)=(x_1,y_1)$ and $B(3,-8)=(x_2,y_2)$

we have,

$(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$=$(-4,6)$

$\Rightarrow \frac{m{x}_2+n{x}_1}{m+n}=-4$

$\Rightarrow m\left(3\right)+n(-6)=-4m-4n$

$\Rightarrow 3m-6n=-4m-4n$

$\Rightarrow 3m+4m=6n-4n$

$\Rightarrow 7m=2n$

$\Rightarrow \frac{m}{n}=\frac{2}{7}$

$\Rightarrow m:n=2:7$