Question

In what ratio is the line segment joining the points $(0,-2,-5)$ and $(-1,3,-6)$ is divided by the $XY$ plane ?Also, find the co-ordinates of the point of division.

Answer 1

Let the line segment formed by joining the points $(0,-2,-5)$ and $(-1,3,-6)$ is divided by the $XY$ plane in the ratio $k:1$ at the point $P$, then the co-ordinates of $P$ are

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

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

$\therefore z-$co-ordinate is zero

$\Rightarrow \frac{-6k-5}{k+1}=0$

$\Rightarrow -6k-5=0$

$\Rightarrow 6k=-5$

$\Rightarrow k=\frac{-5}{6}$

Hence,$XY$ plane divides the line segment $AB$ in the ratio $\frac{-5}{6}:1$

i.e.,$5:6$ externally and the coordinates of the point of division are

$(-\frac{\frac{-5}{6}}{\frac{-5}{6}+1},\frac{3\times \frac{-5}{6}-2}{\frac{-5}{6}+1},0)$

or $(-\frac{\frac{-5}{6}}{\frac{-5+6}{6}},\frac{\frac{-15-12}{6}}{\frac{-5+6}{6}},0)$

or $(5,-27,0)$