Question

If the distance of the point (1, 1, 1) from the origin is half its distance from the plane x + y + z + k = 0, then the values of k are ___________.

Answer 1

Given, distance of the point (1, 1, 1) from the origin is the half its distance from the plane x + y + z + k = 0

Since distance between (1, 1, 1) and (0, 0, 0) $=\sqrt{{\left(1-0\right)}^2+{\left(1-0\right)}^2+{\left(1-0\right)}^2}=\sqrt{3}....\left(1\right)$

Distance of (1, 1, 1) from (x + y + z + k = 0)

$$\begin{gathered} d=\frac{\left|1+1+1+k\right|}{\sqrt{1+1+1}}=\frac{\left|3+k\right|}{\sqrt{3}}....\left(2\right) \\ \mathrm{Since}\left(1\right)=\frac{1}{2}\left(2\right)\mathrm{is}\mathrm{given} \\ \sqrt{3}=\frac{1}{2}\frac{\left|3+k\right|}{\sqrt{3}} \\ \mathrm{i}.\mathrm{e}.2\times 3=\left|3+k\right| \\ \mathrm{i}.\mathrm{e}.\pm 6=3+k \\ \mathrm{i}.\mathrm{e}.k=3\mathrm{or}k=-9 \end{gathered}$$