Determine the ratio in which the point $P (m, 6)$ divides the join of $A (-4, 3)$ and $B (2, 8)$. Also, find the value of $m$.

Given:

$P( m,\ 6)$ divides the line segment joining the points $A( -4,\ 3)$ and $B( 2,\ 8)$.

To do:

We have to find the ratio of division and the value of $m$.

Solution:

Let the point $P( m,\ 6)$ divides the line segment joining the points $A( -4,\ 3)$ and $B(2, 8)$ in the ratio $k:1$.

Using section formula, we have,

$P(x, y)=( \frac{mx_{2}+nx_{1}}{m+n}, \frac{my_{2}+ny_{1}}{m+n})$

This implies,

$P(m, 6)=( \frac{k(2)+1(-4)}{k+1}, \frac{k(8)+1(3)}{k+1})$

$\Rightarrow (m, 6)=( \frac{2k-4}{k+1}, \frac{8k+3}{k+1})$

On comparing,

$\frac{8k+3}{k+1}=6$

$\Rightarrow 8k+3=6( k+1)$

$\Rightarrow 8k+3=6k+6$

$\Rightarrow 8k-6k=6-3$

$\Rightarrow 2k=3$

$\Rightarrow k=\frac{3}{2}$

This implies,

The required ratio is $\frac{3}{2}:1=\frac{3}{2}\times2:1\times2=3:2$.

Therefore,

$m=\frac{2k-4}{k+1}$

$\Rightarrow m=\frac{2(\frac{3}{2})-4}{\frac{3}{2}+1}$

$\Rightarrow m=\frac{3-4}{\frac{3+2\times1}{2}}$

$\Rightarrow m=\frac{-1}{\frac{5}{2}}$

$\Rightarrow m=\frac{-2}{5}$

Therefore, the ratio of division is $3:2$ and the value of $m$ is $\frac{-2}{5}$.

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Updated on: 10-Oct-2022

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