$\begin{array}{1 1}z-y=0 \\ x-2y=0 \\ x+z=0 \\ x-z=0\end{array} $

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- The distance between two points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ is given by $\sqrt {(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$

Let the given points be $A(1,2,3)$ and $B(3,2,-1)$

Let the set of points that are equdistant from $A\:and\:B$ be $P(x,y,z)$

Since $P$ is equdistant from $A$ and $B$,

$\overline {PA}=\overline {PB}$

$\Rightarrow\:(\overline {PA})^2=(\overline {PB})^2$

We know that the distance between two points $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ is given by $\sqrt {(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$

$\Rightarrow (x-1)^2+(y-2)^2+(z-3)^2=(x-3)^2+(y-2)^2+(z+1)^2$

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

$\Rightarrow\:4x-4z=0$

$\Rightarrow\:x-z=0$

$i.e.,$ The required equation is $x-z=0$

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