Find the coordinates of a point equidistant from th four points `O(0,0,0),A(a ,0,0),B(0,b ,0)a n dC(0,0,c)dot`

Options

A. `((a+b+c)/3,(a+b+c)/3,(a+b+c)/3)`
B. (a,b,c)
C. `(a/2,b/2,c/2)`
D. `(a/3,b/3,c/3)`

Answer

C

Explanation

A(0,0,0), B (a,0,0), C(0,b,0), D (0,0,c)
Let the equidistant point be P (x,y,z)
i.e, AP = BP, AP = CP, AP = DP
`rArr sqrt((x-0)^2+(y-0)^2+(z-0)^2)`
`=sqrt((x-a)^2+y^2+z^2)`
`rArr x^2+cancel(y^2)+cancel(z^2)=(x-a)^2+cancel(y^2)+cancel(z^2)`
`rArr x^2=x^2-2ax+a^2`
`rArr a^2-2ax=0`
`rArra(a-2x)=0`
Since `a ne 0,a=2xrArrx=a/2`
Similarly, we will get `y=b/2z=c/2`