Question

The point equidistant from the point $O(0,0,0),A(a,0,0),B(0,b,0)$ and $C(0,0,c)$ has the coordinates

• A. $(a,b,c)$
• B. $(a/2,b/2,c/2)$
• C. $(a/3,b/3,c/3)$
• D. $(a/4,b/4,c/4)$

Answer 1

The correct option is B $(a/2,b/2,c/2)$

$P(x,y,z)$

$PO=\sqrt{x^2+y^2+z^2}$

$PA=\sqrt{(a-x{)}^2+(-y{)}^2+(-z{)}^2}=\sqrt{(a-z{)}^2+y^2+z^2}$

$PB=\sqrt{x^2+(b-y{)}^2+z^2}$

$PC=\sqrt{x^2+y^2+(C-z{)}^2}$

$\Rightarrow PO=PA$

$\sqrt{x^2+y^2+z^2}=\sqrt{(a-x{)}^2+y^2+z^2}$

$x^2+y^2+z^2=(a-x{)}^2+y^2+z^2$

$x^2=(a-x{)}^2$

$x=a-x$

$\Rightarrow x=\frac{a}{2}$

$y=b/2$

$z=c/2$.