Question

Find the ratio in which the point $(-3,k)$ divides the line segment joining the points $(-5,-4)$ and $(-2,3)$. Also find the value of $k$.

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 $m_1:m_2$, then

$(x,y)=(\frac{m_1{x}_2+m_2{x}_1}{m_1+m_2},\frac{m_1{y}_2+m_2{y}_1}{m_1+m_2})$

Given that ratio $m_1:m_2=xy$

points $A(-5,-4)$ and $B(-2,3)$

Let ratio be

$\frac{m_1}{m_2}=\frac{m}{1}$

Therefore,

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

$-3=\frac{m.(-2)+\left(1\right)(-5)}{m+1}$

$-3(m+1)=-2m-5$

$-3m-3=-2m-5$

$-3+5=-2m+3m$

$m=2$

$\frac{m_1}{m_2}=\frac{2}{1}$

Now, $k=\frac{m_1.y_2+m_2.y_1}{m_1+m_2}$

$k=\frac{2.\left(3\right)+1(-4)}{2+1}$

$k=\frac{6-4}{3}$

$k=2/3$

$\therefore k=2/3$