Question

Find the equation of the set of points which are equidistant from the points $(1,2,3)$ and $(3,2,-1)$.

Answer 1

Let $P(x,y,z)$ be the point that is equidistant from points $A(1,2,3)$ and $B(3,2,-1)$.

Accordingly, $PA=PB$

$P{A}^2=P{B}^2$

$(x-1{)}^2+(y-2{)}^2+(z-3{)}^2=(x-3{)}^2+(y-2{)}^2+(z+1{)}^2$

$x^2-2x+1+y^2-4y+4+z^2-6z+9=x^2-6x+9+y^2-4y+4+z^2+2z+1$

$x-2z=0$

Thus, the required equation is $x-2z=0$.