# If $A,B,C$ are three points on $x\:axis,\:y\:axis\:and\:z\:axis$ respectively which are at the distances $a,b,c$ from origin respectively and $P$ is a point which is equidistant from $A,B,C,O$ then the coordinates of $P$ is?

$(a)\;(a,b,c)\qquad(b)\;(\large\frac{a}{2},\frac{b}{2},\frac{c}{2})\qquad(c)\;(\large\frac{a}{3},\frac{b}{3},\frac{c}{3})\qquad(d)\;(2a,2b,2c)$

Given: $A(a,0,0),\:B(0,b,0)\:and\:C(0,0,c)$
Let $P(x,y,z)$ be equidistant from $O,A,B\:and \:C$
$\Rightarrow\: OP=AP=BP=CP$
$\Rightarrow\:x^2+y^2+z^2=(x-a)^2+y^2+z^2=x^2+(y-b)^2+z^2=x^2+y^2+(z-c)^2$
$\Rightarrow\:x=\large\frac{a}{2}$, $y=\large\frac{b}{2}$ and $z=\large\frac{c}{2}$
$\therefore$ The point $P$ is $(\large\frac{a}{2},\frac{b}{2},\frac{c}{2})$