Question

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

Answer 1

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

$(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

Point $P(9a-2,-b)$ divides the line segment joining the points $A(3a+1,-3)$ and $B(8a,5)$ in the ratio $3:1$ . But , the coordinates of $P$ are $(9a-2,-b)$ .

Using section formula , we have

$9a-2=\frac{3\left(8a\right)+1(3a+1)}{3+1}$

$=\frac{24a+3a+1}{4}$

$\Rightarrow 36a-8=27a+1$

$\Rightarrow 36a-27a=8+1$

$\Rightarrow 9a=9$

$\Rightarrow a=\frac{9}{9}=1$

And

$-b=\frac{3(+5)+1(-3)}{3+1}$

$\Rightarrow -b=\frac{+15-3}{4}=\frac{12}{4}$

$\Rightarrow b=-3$

Hence , $a=+1$ and $b=-3$