Question

Find the coordinates of a point equidistant from the four points $O(0,0,0)$, $A(l,0,0)$, $B(0,m,0)$ and $C(0,0,n)$.

• A. $(l,m,n)$
• B. $\left(\frac{l}{2},\frac{m}{2},\frac{n}{2}\right)$
• C. $\left(\frac{l}{2},m,\frac{n}{2}\right)$
• D. $\left(2l,2m,2n\right)$

Answer 1

Answer: (B)

Let $P(x,y,z)$ be the required point.
Then $OP=PA=PB=PC$.
Now $OP=PA$
$\Rightarrow O{P}^2=P{A}^2$
$\Rightarrow x^2+y^2+z^2={\left(x-l\right)}^2+{\left(y-0\right)}^2+{\left(z-0\right)}^2$
$\Rightarrow x=\frac{l}{2}$
Similarly, $OP=PB$
$\Rightarrow y=\frac{m}{2}$ and $OP=PC$
$\Rightarrow z=\frac{n}{2}$
Hence, the coordinates of the required point are
$\left(\frac{l}{2},\frac{m}{2},\frac{n}{2}\right)$.