Question

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

Answer 1

Let the plane x + y + z = 5 divides the line joining points (2, -1, 3) and (-1, 2, 1) in the ratio k : 1.

So, the required coordinates are

$(\frac{-k+2}{k+1},\frac{2k-1}{k+1},\frac{k+3}{k+1})$

Since, this point lies on the plane x + y + z = 5

$\therefore \frac{-k+2}{k+1}+\frac{2k-1}{k+1}+\frac{k+3}{k+1}=5$

$\Rightarrow -k+2+2k-1+k+3=5(k+1)$

$\Rightarrow 2k+4=5k+5$

$\Rightarrow 3k=-1$

$\therefore k=-\frac{1}{3}$

So, the required ratio is $\frac{-1}{3}:1or1:3$ externally divides the plane.

The coordinates of point of division are $(\frac{5}{2},\frac{-5}{2},4).$