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)`
B. `(a//2,b//2,c//2)`
C. `(a//3,b//3,c//3)`
D. `(a//4,b//4,c//4)`

Answer

B

Explanation

Let `P(x,y,z)` be the required point. Then
`OP=AP=BP=CP`
Now `OP=AP`
`impliesOP^(2)=AP^(2)`
`impliesx^(2)+y^(2)+z^(2)=(x-a)^(2)+y^(2)+z^(2)`
`implies-2ax+a^(2)=0impliesx=a/2`
Similarly, `OP=BP` and `OP=CP`implies `y=b/2` and `z=c/2`
Hence required point has the coordinates `(a//2,b//2,c//2)`.