Find the ratio in which the point $P (-1, y)$ lying on the line segment joining $A (-3, 10)$ and $B (6, -8)$ divides it. Also find the value of $y$.

Given:

Point $P(-1,\ y)$ divides the line segment joining the points $A( -3,\ 10)$ and $B( 6,\ -8)$.

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,

$x_1=-3, y_1=10, x_2=6, y_2=-8$

Let the ratio be $m:n$

This implies,

$P(-1,y)=( \frac{m(6)+n(-3)}{m+n},\ \frac{m(-8)+n(10)}{m+n})$

Therefore, equating coordinates on both sides, we get,

$\frac{6m-3n}{m+n}=-1$

$\Rightarrow  6m-3n=-1(m+n)$

$\Rightarrow  6m-3n=-m-n)$

$\Rightarrow  6m+m=3n-n$

$\Rightarrow  7m=2n$

$\Rightarrow  \frac{m}{n}=\frac{2}{7}$

​$\Rightarrow  m:n=2:7$

Now,

$y=\frac{-8m+10n}{m+n}$

​$\Rightarrow  y=\frac{-8( 2)+10( 7)}{ 2+7}$

$\Rightarrow  9(y)=-16+70$

$\Rightarrow  9y=54$

$\Rightarrow y=\frac{54}{9}=6$

 The required ratio is $2:7$ and the value of $y$ is $6$. 

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Updated on: 10-Oct-2022

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