In what ratio does the point $( \frac{24}{11} ,\ y)$ divide the line segment joining the points $P( 2,\ -2)$ and $( 3,\ 7)$ ?Also find the value of $y$. 

Let us say that the given point divides the line segment  into $k:1$ ratio,

 Using section formula, we have point of division, $ ( x,\ y) =(\frac{nx_{1} +mx_{2}}{m+n} ,\ \frac{ny_{1} +my_{2}}{m+n})$

$\Rightarrow ( \frac{24}{11} ,\ y) =(\frac{1\times2 +k\times3}{k+1} ,\ \frac{1\times-2 +k\times7}{k+1})$

$\Rightarrow ( \frac{24}{11} ,\ y) =(\frac{2+3k}{k+1} ,\ \frac{7k-2}{k+1})$

On comparing,

$\Rightarrow\frac{24}{11}=\frac{2+3k}{k+1}    \ \ \ \ \ \ ...........( 1)$

And $y=\frac{7k-2}{k+1}          \ \ \ \ \ \ ........... ( 2)$

$\Rightarrow 24( k+1)=11( 2+3k)$

$\Rightarrow 24k+24=22+33k$

$\Rightarrow 33k-24k=24-22$

$\Rightarrow 9k=2$

$\Rightarrow k=\frac{2}{9}$

On subtituting this value in $( 2)$

 $\Rightarrow y=\frac{7k-2}{k+1}=\frac{7( \frac{2}{9})-2}{( \frac{2}{9})+1}$

$\Rightarrow y=\frac{14-18}{2+9}$

$\Rightarrow y=\frac{-4}{11}$

Hence, the given point divides the line segment in  $2:9$ ratio and $y=\frac{-4}{11}$.

Tutorialspoint

Simply Easy Learning

Updated on: 10-Oct-2022

6K+ Views

Kickstart Your Career

Get certified by completing the course

Get Started