Question

$A,B,C$ are three points on the axes of $x,y$ and $z$ respectively at distance $a,b,c$ from the origin $O$; then the co - ordinates of the point which is equidistant from $A,B,C$ and $O$ is

• A. $(a,b,c)$
• B. $(\frac{a}{2},\frac{b}{2},\frac{c}{2})$
• C. $(\frac{a}{3},\frac{b}{3},\frac{c}{3})$
• D. None of these

Answer 1

Answer: (B)

$(\frac{a}{2},\frac{b}{2},\frac{c}{2})$

Let $P$ be the required point $(x,y,z)$ and the point
$A,B,C$ and $O$ are $(a,0,0),(0,b,0),(0,0,c)$ and $(0,0,0)$ ;

We are given that $PO=PA=PB=PC.$
Taking $PO=PA$ or $P{O}^2=P{A}^2,$ we get
$x^2+y^2+z^2={(x-a)}^2+y^2+z^2$
$0=a^2-2ax$ i.e. $x=\frac{a}{2}$
Similarly taking $P{O}^2=P{B}^2$ and $P{O}^2=P{C}^2,$ we get

$y=\frac{b}{2}$ and $z=\frac{c}{2}$