# The mid-point $P$ of the line segment joining the points $A (-10, 4)$ and $B (-2, 0)$ lies on the line segment joining the points $C (-9, -4)$ and $D (-4, y)$. Find the ratio in which $P$ divides $CD$. Also, find the value of $y$.

Given:

The mid-point $P$ of the line segment joining the points $A (-10, 4)$ and $B (-2, 0)$ lies on the line segment joining the points $C (-9, -4)$ and $D (-4, y)$.

To do:

We have to find the ratio in which $P$ divides $CD$ and the value of $y$.

Solution:

Let $P$ divides $C(9,-4)$ and $D(-4, y)$ in the ratio $m: n$

P is the mid-point of the line segment joining the points $A (-10, 4)$ and $B (-2, 0)$

This implies,

$P=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$

$=\left(\frac{-10-2}{2}, \frac{4+0}{2}\right)$

$=\left(\frac{-12}{2}, \frac{4}{2}\right)$
$=(-6,2)$
P lies on CD.

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

$(x,\ y)=( \frac{mx_2+nx_1}{m+n},\ \frac{my_2+ny_1}{m+n})$

This implies,

$(-6,2)=(\frac{m(-4)+n(-9)}{m+n},\frac{3 \times y+2 \times(-4)}{3+2})$

On comparing, we get,

$-6=\frac{m(-4)+n(-9)}{m+n}$

$\Rightarrow -6 m-6n=-4 m-9n$

$\Rightarrow -6 m+4 m=-9n+6n$

$\Rightarrow -2 m=-3 n$

$\Rightarrow \frac{m}{n}=\frac{-3}{-2}$

$=\frac{3}{2}$

$2=\frac{3 \times y+2 \times(-4)}{3+2}$

$\Rightarrow 2=\frac{3 y-8}{5}$

$\Rightarrow 10=3 y-8$

$\Rightarrow 3 y=10+8$

$\Rightarrow 3 y=18$

$\Rightarrow y=\frac{18}{3}$

$\Rightarrow y=6$

The ratio in which $P$ divides $CD$ is $3:2$ and the value of $y$ is $6$.

Tutorialspoint

Simply Easy Learning

Updated on: 10-Oct-2022

42 Views