Find the ratio in which the line segment joining
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.

Solution:

The given points are : A(1, – 5) and B(–4, 5). Let the required ratio = k : 1 and the required point be P(x, y)

Part-I : To find the ratio

Since, the point P lies on x-axis,

$\therefore$ Its $y$-coordinate is 0 .

$x=\frac{m_{1} x_{2}+m_{2} x_{1}}{m_{1}+m_{2}}$ and $0=\frac{m_{1} y_{2}+m_{2} y_{1}}{m_{1}+m_{2}}$

$\Rightarrow x=\frac{-4 k+1}{\mathbf{k}+1}$ and $0=\frac{\mathbf{5 k}-\mathbf{5}}{\mathbf{k}+1}$

$\Rightarrow x(k+1)=-4 k+1$

and $5 \mathrm{k}-5=0 \Rightarrow \mathrm{k}=1$

$\Rightarrow \quad x(k+1)=-4 k+1$

$\Rightarrow x(1+1)=-4+1 \quad[\because k=1]$

$\Rightarrow 2 x=-3$

$\Rightarrow x=-\frac{3}{2}$

$\therefore$ The required ratio $\mathrm{k}: 1=1: 1$

Coordinates of $\mathrm{P}$ are $(\mathrm{x}, 0)=\left(\frac{-\mathbf{3}}{\mathbf{2}}, \mathbf{0}\right)$
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