Question

In what ratio is the line segment joining the points $(2,-4,-7)$ and $(-3,5,-8)$ 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 $(2,-4,-7)$ and $(-3,5,-8)$ is divided by the $XY$ 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 $XY$ plane,

$\therefore z-$co-ordinate is zero

$\Rightarrow \frac{-8k-7}{k+1}=0$

$\Rightarrow -8k-7=0$

$\Rightarrow 8k=-7$

$\Rightarrow k=\frac{-7}{8}$

Hence,$XY$ plane divides the line segment $AB$ in the ratio $\frac{-7}{8}:1$

i.e.,$7:8$ externally and the coordinates of the point of division are

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

or $(\frac{\frac{21+16}{8}}{\frac{-7+8}{8}},\frac{\frac{-35-32}{8}}{\frac{-7+8}{8}},\frac{\frac{56-56}{8}}{\frac{-7}{8}+1})$

or $(\frac{37}{1},\frac{-67}{1},0)$

or $(37,-67,0)$