Question

If the x-coordinate of a point P on the join of Q (2, 2, 1) and R (5, 1, −2) is 4, then its z-coordinate is
(a) 2
(b) 1
(c) −1
(d) −2

Answer 1

$$\begin{gathered} \left(\mathrm{c}\right)-1 \\ \mathrm{Suppose}\mathrm{the}\mathrm{point}P\mathit{}\mathrm{divides}\mathrm{the}\mathrm{line}\mathrm{joining}\mathrm{the}\mathrm{point}Q\mathit{}\left(2,2,1\right)\mathit{}\mathrm{and}R\mathit{}\left(5,1,-2\right)\mathit{}\mathrm{in}\mathrm{the}\mathrm{ratio}k:1. \\ \mathrm{Using}\mathrm{the}\mathrm{section}\mathrm{formula},\mathrm{the}\mathrm{coordinates}\mathrm{of}\mathrm{the}\mathrm{point}\mathrm{of}\mathrm{intersection}\mathrm{are}\mathrm{given}\mathrm{by} \\ \left(\frac{k\left(5\right)+2}{k+1},\frac{k\left(\mathit{1}\right)+2}{k+1},\frac{k\left(\mathit{-}\mathit{2}\right)+1}{k+1}\right) \\ \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\mathrm{the}X-\mathrm{coordinate}\mathrm{of}P\mathit{}\mathrm{is}4. \\ \Rightarrow \frac{k\left(5\right)+2}{k+1}=4 \\ \Rightarrow 5k+2=4\left(k+1\right) \\ \Rightarrow k=2 \\ \mathrm{Now}, \\ \mathrm{Z}-\mathrm{coordinate}\mathrm{of}\mathrm{P}=\frac{\mathrm{k}\left(\mathit{-}\mathit{2}\right)+1}{\mathrm{k}+1} \\ =\frac{2\left(-2\right)+1}{2+1}\left[\mathrm{Substituting}\mathit{}k=2\right] \\ =-1 \end{gathered}$$