Question

In what ratio is the line segment joining $(0,2)$ and $(6,8)$ divided by the line $2x+2y=5?$

Answer 1

The given line is $2x+2y=5$ ..........$\left(1\right)$

Let the line $\left(1\right)$ divide the line segment joining the point $(0,2)$ and $(6,8)$ at the point $P$ in the ratio $k:1$ then the co-ordinates of point $P$ are

$(\frac{6k-0}{k+1},\frac{8k+2}{k+1})$

Since $P$ lies on $\left(1\right)$, we get $2\frac{6k}{k+1}+2\frac{8k+2}{k+1}=5$

$\Rightarrow 12k+16k+4=5k+5$

$\Rightarrow 28k-5k=5-4$

$\Rightarrow 23k=1$

$\Rightarrow k=\frac{1}{23}$

Hence, the required ratio is $\frac{1}{23}:1$ or $1:23$ internally.