Question

Find the ratio in which the line segment joining $A(1,-5)$ and $B(-4,5)$ is divided by the $X-$axis. Also find the coordinates of the point of division.

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 $k:1$ and the point on the X-axis be $(x,0)$

Let $(x_1,y_1)=(1,-5),(x_2,y_2)=(-4,5)$

$\Rightarrow y=\frac{k{y}_2+y_1}{k+1}$

$\Rightarrow 0=\frac{k\times 5+(-5)}{k+1}$

$\Rightarrow 0=\frac{5k-5}{k+1}$

$\Rightarrow 0=5k-5$

$\Rightarrow -5k=-5$

$\Rightarrow k=\frac{-5}{-5}$

$\therefore$Required ratio $(m_1:m_2)=1:1$

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

$\Rightarrow x=\frac{1\times (-4)+1\times 1}{1+1}$

$\Rightarrow x=\frac{-4+1}{2}=\frac{-3}{2}$

$\therefore$The ratio $=1:1$ and the required point of intersection $=(\frac{-3}{2},0)$