If $ \mathrm{P}(9 a-2,-b) $ divides line segment joining $ \mathrm{A}(3 a+1,-3) $ and $ \mathrm{B}(8 a, 5) $ in the ratio $ 3: 1 $, find the values of $ a $ and $ b $.

Given: 

$P (9a – 2, -b)$ divides the line segment joining $A (3a + 1, -3)$ and $B (8a, 5)$ in the ratio $3 : 1$.

To do: 

We have to find the values of $a$ and $b$.

Solution:

Using the division formula,

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

Here,

$x_1=3a+1,\ y_1=-3,\ x_2=8a,\ y_2=5,\ x=9a-2,\ y=-b, \ m=3$ and $n=1$.

$( 9a-2,\ -b)=( \frac{3\times(8a)+1\times(3a+1)}{3+1},\ \frac{3\times5+1\times(-3)}{3+1})$

$(9a-2,\ -b)=( \frac{24a+3a+1}{4},\ \frac{15-3}{4})$

$( 9a-2,\ -b)=( \frac{27a+1}{4},\ \frac{12}{4})$

This implies,

$9a-2=\frac{27a+1}{4}$ and $-b=3$

$4(9a-2)=27a+1$ and $b=-3$

$36a-8=27a+1$

$36a-27a=8+1$

$9a=9$

$a=1$

Therefore, the values of $a$ and $b$ are $1$ and $-3$ respectively. 

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Updated on: 10-Oct-2022

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