Question

In what ratio is the line segment joining the points $(1,1,4)$ and $(2,-2,5)$ 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 $(1,1,4)$ and $(2,-2,5)$ is divided by the $YZ$ plane in the ratio $k:1$ at the point $P$, then the co-ordinates of $P$ are

$(\frac{2k+1}{k+1},\frac{-2k+1}{k+1},\frac{5k+4}{k+1})$

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

$\therefore x-$co-ordinate is zero

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

$\Rightarrow\,2k+1=0

$\Rightarrow k=\frac{-1}{2}$

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

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

$(\frac{2\times \frac{-1}{2}+1}{\frac{-1}{2}+1},\frac{-2\times \frac{-1}{2}+1}{\frac{-1}{2}+1},\frac{5\times \frac{-1}{2}+4}{\frac{-1}{2}+1})$

or $(\frac{-1+1}{\frac{-1+2}{2}},\frac{1+1}{\frac{-1+2}{2}},\frac{\frac{-5+8}{2}}{\frac{-1+2}{2}})$

or $(0,4,3)$