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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