Question

Find the ratio in which the line segment joining the points, (2, -1, 3) and (-1, 2, 1) is divided by the plane $x+y+z=5$.

Answer 1

(2, -1, 3) and (-1, 2, 1)

$x+y+z=5$

Assume plane divides line in ratio $\lambda :1$

so point P which is dividing line in $\lambda :1$ ratio is

$P=(\frac{-\lambda +2}{\lambda +1},\frac{2\lambda -1}{\lambda +1},\frac{\lambda +3}{\lambda +1})$

P lies on plane $x+y+z=5$

$-\lambda +2+2\lambda -1+\lambda +3=5\lambda +5$

$3\lambda =-1\Rightarrow \lambda =-1:3$

So plane dividing line in 1 : 3 ratio externally