Find the ratio in which point $P( x,\ 2)$ divides the line segment joining the points $A( 12,\ 5)$ and $B( 4,\ −3)$. Also find the value of $x$.

Given: Points $P( x,\ 2)$ divides the line segment joining the points $A( 12,\ 5)$ and $B( 4,\ −3)$.

To do: To find the ratio of division and also to find the value of $x$.

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 

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

Let the ratio be $m:n$

Then, $\frac{m( -3)+n( 5)}{m+n}=2$

$\Rightarrow  -3m+5n=2m+2n$

$\Rightarrow  -5m=-3n$

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

​$\Rightarrow  m:n=3:5$

Now, $x=\frac{mx_2+nx_1}{m+n}$

​$\Rightarrow  x=\frac{3( 4)+5( 12)}{ 3+5}$

$\Rightarrow  8x=12+60$

$\Rightarrow  8x=72$

$\Rightarrow x=\frac{72}{8}=9$
 

Updated on: 10-Oct-2022

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