Question

Find the ratio in which the line segment joining the points $(-3,10)$ and $(6,-8)$ is divided by $(-1,6)$.

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 $m_1:m_2$

Take $(x_1,y_1)=(-3,10);(x_2,y_2)=(6,-8)$ and $(x,y)=(-1,6)$

$\therefore x=\frac{m_1{x}_2+m_2{x}_1}{m_1+m_2}$

$\Rightarrow -1=\frac{m_1\times 6+m_2\times -3}{m_1+m_2}$

$\Rightarrow -m_1-m_2=6m-3m_2$

$\Rightarrow -m_1-6m_1=-3m_2+m_2$

$\Rightarrow -7m_1=-2m_2$

$\Rightarrow \frac{m_1}{m_2}=\frac{2}{7}$

$\therefore$The required ratio is $2:7$.