Question

Question 12
If P(9a-2,-b) divides line segment joining A(3a+1,-3) and B(8a,5) in the ratio 3:1, then find the values of a and b.

Answer 1

LetP(9a-2,-b) divides AB internally in the ratio 3:1.
By section formula,
$9a-2=\frac{3\left(8a\right)+1(3a+1)}{3+1}$ and $-b=\frac{3\left(5\right)+1(-3)}{3+1}$
[ By internal section formula, the coordinates of point P divides the ling segment joining the point $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m_1:m_2$ internally is
$(\frac{m_2{x}_1+m_1{x}_2}{m_1+m_2},\frac{m_2{y}_1+m_1{y}_2}{m_1+m_2})]\Rightarrow 9a-2=\frac{24a+3a+1}{4}and-b=\frac{15-3}{4}\Rightarrow 9a-2=\frac{27a+1}{4}and-b=\frac{12}{4}\Rightarrow 36a-8=27a+1andb=-3\Rightarrow 36a-27a-8-1=0\Rightarrow 9a-9=0\therefore a=1\text{Hence, the required values of a and b are 1 and -3.}$