Determine the ratio in which the point $(-6, a)$ divides the join of $A (-3, 1)$ and $B (-8, 9)$. Also find the value of $a$.

Given:

Point $( -6,\ a)$ divides the line segment joining the points $A( -3,\ 1)$ and $B( -8,\ 9)$.

To do:

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

Solution:

Let the point $P( -6,\ a)$ divides the line segment joining the points $A( -3,\ 1)$ and $B(-8, 9)$ 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,

$P(-6, a)=( \frac{k(-8)+1(-3)}{k+1}, \frac{k(9)+1(1)}{k+1})$

$\Rightarrow (-6, a)=( \frac{-8k-3}{k+1}, \frac{9k+1}{k+1})$

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,

$a=\frac{9k+1}{k+1}$

$\Rightarrow a=\frac{9(\frac{3}{2})+1}{\frac{3}{2}+1}$

$\Rightarrow a=\frac{\frac{27}{2}+1}{\frac{3+2\times1}{2}}$

$\Rightarrow a=\frac{\frac{27+2(1)}{2}}{\frac{5}{2}}$

$\Rightarrow a=\frac{29}{5}$

Therefore, the ratio of division is $3:2$ and the value of $a$ is $\frac{29}{5}$.

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Updated on: 10-Oct-2022

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