Question

Find the coordinates of a point which equidistant from the four points $o(0,0,0),A(l,0,0),B(0,m,0)andC(0,0,n)$.

Answer 1

let P(x,y,z) be required point. Then , $OP=PA=PB=PC.$

Now , $OP=PA\Rightarrow O{P}^2=P{A}^2$

$\Rightarrow x^2+y^2+z^2=(x-l{)}^2+(y-0{)}^2+(z-0{)}^2$

$[\because distencebetweentwopoints=\sqrt{(x^2-x_1{)}^2+(y_2-y_1{)}^2+(x_2-z_1{)}^2}]$

$\Rightarrow x^2+y^2+x^2=x^2+l^2-2lx+y^2+z^2$

$\Rightarrow l^2-2lk=0\Rightarrow x=\frac{1}{2}[\because l\ne 0]$

Sinmilarly,$0P=PB\Rightarrow y=\frac{m}{2}$

and ,$OP=PC\Rightarrow z=\frac{n}{2}$

Hence, the coordinates of the required point are $(\frac{l}{2},\frac{m}{2},\frac{n}{2})$