Question

Find the points on z-axis which are at a distance from the point (1, 2, 3).

Answer 1

Let the point be A (0, 0, z).Then,

AP = $\sqrt{21}$
$$\begin{gathered} \Rightarrow \sqrt{{\left(0-1\right)}^2+{\left(0-2\right)}^2+{\left(z-3\right)}^2}=\sqrt{21} \\ \Rightarrow {\left(-1\right)}^2+{\left(-2\right)}^2+{\left(z-3\right)}^2=21 \\ \Rightarrow 1+4+{\left(z-3\right)}^2=21 \\ \Rightarrow {\left(z-3\right)}^2=21-5 \\ \Rightarrow {\left(z-3\right)}^2=16 \\ \Rightarrow z-3=\pm 4 \\ \Rightarrow z-3=4\mathrm{or}z-3=-4 \\ \Rightarrow z=7\mathrm{or}z=-1 \end{gathered}$$
Hence, the coordinates of the required point are (0, 0, 7) and (0, 0, −1).