Question

If $x$ coordinates of a point $P$ of the line joining the points $Q(2,2,1)\mathrm{and}R(5,2,-2)$ is $4$, then the $z$-co-ordinates of $P$ is:

• A. $-2$
• B. $-1$
• C. $1$
• D. $2$

Answer 1

The correct option is B

$-1$

Explanation for the correct option.

The given points are $Q(2,2,1)\mathrm{and}R(5,2,-2)$

Equation of the line passing through the points $Q\mathrm{and}R$ will be:

$\begin{array}{rcl}\frac{x-2}{3}& =& \frac{y-2}{0}=\frac{z-1}{-3}=\lambda \left[\mathit{B}\mathit{y}\mathbf{}\mathit{E}\mathit{q}\mathit{u}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{l}\mathit{i}\mathit{n}\mathit{e}\mathbf{}\mathit{i}\mathit{s}\mathbf{}\frac{\mathbf{x}\mathbf{-}{\mathbf{x}}_{\mathbf{1}}}{{\mathbf{x}}_{\mathbf{2}}\mathbf{-}{\mathbf{x}}_{\mathbf{1}}}\mathbf{=}\frac{\mathbf{y}\mathbf{-}{\mathbf{y}}_{\mathbf{1}}}{{\mathbf{y}}_{\mathbf{2}}\mathbf{-}{\mathbf{y}}_{\mathbf{1}}}\mathbf{=}\frac{\mathbf{z}\mathbf{-}{\mathbf{z}}_{\mathbf{1}}}{{\mathbf{z}}_{\mathbf{2}}\mathbf{-}{\mathbf{z}}_{\mathbf{1}}}\right]\\ & \Rightarrow & \frac{x-2}{3}=\lambda \\ \Rightarrow x& =& 3\lambda +2\\ \Rightarrow 4& =& 3\lambda +2\\ \Rightarrow \lambda & =& \frac{2}{3}\end{array}$

Now, the $z$-co-ordinate will be

$\begin{array}{rcl}\frac{z-1}{-3}& =& \lambda \\ & \Rightarrow & \frac{z-1}{-3}=\frac{2}{3}\\ \Rightarrow z& =& -1\end{array}$

Hence, option B is correct.