Find the ratio in which the points $(2, y)$ divides the line segment joining the points $A (-2, 2)$ and $B (3, 7)$. Also, find the value of $y$.


Given:

Points $(2,\ y)$ divides the line segment joining the points $A( -2,\ 2)$ and $B( 3,\ 7)$.

To do:

We have to find the ratio of division and the value of $y$.

Solution:

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 

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

Here,

$x_1=-2, y_1=2, x_2=3, y_2=7$

Let the ratio be $m:n$

This implies,

$(2,y)=( \frac{m(3)+n(-2)}{m+n},\ \frac{m(7)+n(2)}{m+n})$

Therefore, equating coordinates on both sides, we get,

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

$\Rightarrow  3m-2n=2m+2n$

$\Rightarrow  3m-2m=2n+2n$

$\Rightarrow  m=4n$

$\Rightarrow  \frac{m}{n}=\frac{4}{1}$

​$\Rightarrow  m:n=4:1$

Now,

$y=\frac{7m+2n}{m+n}$

​$\Rightarrow  y=\frac{7( 4)+2( 1)}{ 4+1}$

$\Rightarrow  5(y)=28+2$

$\Rightarrow  5y=30$

$\Rightarrow y=\frac{30}{5}=6$

The required ratio is $4:1$ and the value of $y$ is $6$.

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Updated on: 10-Oct-2022

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