Question

In what ratio is the line segment joining the points $(-2,4,7)$ and $(3,-5,8)$ is divided by the $XZ$ plane ?Also, find the co-ordinates of the point of division.

Answer 1

Let the line segment formed by joining the points $(-2,4,7)$ and $(3,-5,8)$ is divided by the $XZ$ plane in the ratio $k:1$ at the point $P$, then the co-ordinates of $P$ are

$(\frac{3k-2}{k+1},\frac{-5k+4}{k+1},\frac{8k+7}{k+1})$

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

$\therefore y-$co-ordinate is zero

$\Rightarrow \frac{-5k+4}{k+1}=0$

$\Rightarrow -5k+4=0$

$\Rightarrow 5k=4$

$\Rightarrow k=\frac{4}{5}$

Hence,$XZ$ plane divides the line segment $AB$ in the ratio $\frac{4}{5}:1$

i.e.,$4:5$ internally and the coordinates of the point of division are

$(\frac{3\times \frac{4}{5}-2}{\frac{4}{5}+1},\frac{-5\times \frac{4}{5}+4}{\frac{4}{5}+1},\frac{8\times \frac{4}{5}+7}{\frac{4}{5}+1})$

or $(\frac{\frac{12-10}{5}}{\frac{4+5}{5}},\frac{-4+4}{\frac{4}{5}+1},\frac{\frac{32}{5}+7}{\frac{4+5}{5}})$

or $(\frac{\frac{2}{5}}{\frac{9}{5}},0,\frac{\frac{32+35}{5}}{\frac{9}{5}})$

or $(\frac{2}{9},0,\frac{67}{9})$