Question

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

$\therefore x-$co-ordinate is zero

$\Rightarrow \frac{3k-2}{k+1}=0$

$\Rightarrow 3k-2=0$

$\Rightarrow 3k=2$

$\Rightarrow k=\frac{2}{3}$

Hence,$YZ$ plane divides the line segment $AB$ in the ratio $\frac{2}{3}:1$

i.e.,$2:3$ internally and the coordinates of the point of division are

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

or $(\frac{2-2}{\frac{2+3}{3}},\frac{\frac{-10+12}{3}}{\frac{2+3}{3}},\frac{\frac{16+21}{3}}{\frac{2+3}{3}})$

or $(0,\frac{\frac{2}{3}}{\frac{5}{3}},\frac{\frac{37}{3}}{\frac{5}{3}})$

or $(0,\frac{2}{5},\frac{37}{5})$