Question

Write the ratio in which the line segment joining (a, b, c) and (−a, −c, −b) is divided by the xy-plane.

Answer 1

$$\begin{gathered} \mathrm{Suppose}\mathrm{the}\mathrm{line}\mathrm{segment}\mathrm{joining}\mathrm{the}\mathrm{points}\mathit{}\left(a\mathit{,}\mathit{}b\mathit{,}\mathit{}c\right)\mathrm{and}\mathit{}\left(\mathit{-}a\mathit{,}\mathit{}\mathit{-}c\mathit{,}\mathit{}\mathit{-}b\right)\mathrm{is}\mathrm{divided}\mathrm{by}\mathrm{the}XY-\mathrm{plane}\mathrm{at}\mathrm{a}\mathrm{point}\mathit{}R\mathrm{in}\mathrm{the}\mathrm{ratio}\mathrm{\lambda }:1. \\ \mathrm{Coordinates}\mathrm{of}R\mathrm{are} \\ \left(\frac{\lambda \left(\mathit{-}a\right)\mathit{+}\mathit{1}\left(a\right)}{\lambda \mathit{+}\mathit{1}}\mathit{,}\mathit{}\frac{\lambda \left(\mathit{-}c\right)\mathit{+}\mathit{1}\left(b\right)}{\lambda \mathit{+}\mathit{1}}\mathit{,}\mathit{}\frac{\lambda \left(\mathit{-}b\right)\mathit{+}\mathit{1}\left(c\right)}{\lambda \mathit{+}\mathit{1}}\right) \\ \mathrm{Since}\mathit{}R\mathrm{lies}\mathrm{on}\mathit{}XY-\mathrm{plane},\mathit{}Z-\mathrm{coordinate}\mathrm{of}R\mathrm{must}\mathrm{be}\mathrm{zero}. \\ \Rightarrow \frac{\mathrm{\lambda }\left(\mathit{-}\mathrm{b}\right)\mathit{+}\mathit{1}\left(\mathrm{c}\right)}{\mathrm{\lambda }\mathit{+}\mathit{1}}=0=\frac{\mathrm{c}}{\mathrm{b}} \\ \mathrm{Thus},\mathrm{the}\mathrm{required}\mathrm{ratio}\mathrm{is}\raisebox{1ex}{$\mathrm{c}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{b}:1\mathrm{or}\mathrm{c}:\mathrm{b}$}\right.. \\ \mathrm{Hence},\mathrm{the}XY-\mathrm{plane}\mathrm{divides}\mathrm{the}\mathrm{line}\mathrm{in}\mathrm{the}\mathrm{ratio}\mathrm{c}:\mathrm{b}. \end{gathered}$$