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