Question

In what ratio is the line segment joining the points $(2,2,5)$ and $(1,-3,6)$ is divided by the $YZ$ plane ?Also, find the co-ordinates of the point of division.

Answer 1

Let the line segment formed by joining the points $(2,2,5)$ and $(1,-3,6)$ is divided by the $YZ$ plane in the ratio $k:1$ at the point $P$, then the co-ordinates of $P$ are

$(\frac{k+2}{k+1},\frac{-3k+2}{k+1},\frac{6k+5}{k+1})$

But $P$ lies in the $YZ$ plane,

$\therefore x-$co-ordinate is zero

$\Rightarrow \frac{k+2}{k+1}=0$

$\Rightarrow k+2=0$

$\Rightarrow k=-2$

Hence,$YZ$ plane divides the line segment $AB$ in the ratio $-2:1$

i.e.,$2:1$ externally and the coordinates of the point of division are

$(\frac{-2+2}{k+1},\frac{-3\times -2+2}{-2+1},\frac{6\times -2+5}{-2+1})$

or $(0,\frac{6+2}{-1},\frac{-12+5}{-1})$

or $(0,-8,7)$