Question

The line $y-x+2=0$ divides the line joining $(3,-1)$ and $(8,9)$ in the ratio $\lambda :1$, where $\lambda$ is

• A. $\frac{2}{3}$
• B. $\frac{5}{3}$
• C. $\frac{5}{2}$
• D. $\frac{3}{2}$

Answer 1

Answer: (A)

Suppose the line $y-x+2=0$ divides the line segment joining $A(3,-1)$ and $B(8,9)$ in the ratio $\lambda :$ 1at point $P$, then the coordinates of the point $P$ are $\left(\frac{8\lambda +3}{\lambda +1},\frac{9\lambda -1}{\lambda +1}\right)$.
But $P$ lies on $y-x+2=0$, therefore
$\left(\frac{9\lambda -1}{\lambda +1}\right)-\left(\frac{8\lambda +3}{\lambda +1}\right)+2=0$
$\Rightarrow 9\lambda -1-8\lambda -3+2\lambda +2=0$
$\Rightarrow 3\lambda -2=0$
or $\lambda =\frac{2}{3}$
So, the required ratio is $\frac{2}{3}:1$, i.e., $2:3$ (internally), since here $\lambda$ is positive.