Find the ratio in which the 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:

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

To do:

We have to find the ratio of division and 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 $m:n$

This implies,

$P(x,\ 2)=( \frac{m(4)+n(12)}{m+n},\ \frac{m(-3)+n(5)}{m+n})$

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

$\Rightarrow 2(m+n)=-3m+5n$

$\Rightarrow  2m+2n=-3m+5n$

$\Rightarrow  2m+3m=5n-2n$

$\Rightarrow  5m=3n$

$\Rightarrow  \frac{m}{n}=\frac{3}{5}$

​$\Rightarrow  m:n=3:5$

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

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

$\Rightarrow  8x=12+60$

$\Rightarrow  8x=72$

$\Rightarrow x=\frac{72}{8}=9$

 The required ratio is $3:5$ and the value of $x$ is $9$.

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Updated on: 10-Oct-2022

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