Question:
Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Solution:
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
$\Rightarrow \mathrm{PA}^{2}=\mathrm{PB}^{2}$
$\Rightarrow(x-1)^{2}+(y-2)^{2}+(z-3)^{2}=(x-3)^{2}+(y-2)^{2}+(z+1)^{2}$
$\Rightarrow x^{2}-2 x+1+y^{2}-4 y+4+z^{2}-6 z+9=x^{2}-6 x+9+y^{2}-4 y+4+z^{2}+2 z+1$
$\Rightarrow-2 x-4 y-6 z+14=-6 x-4 y+2 z+14$
$\Rightarrow-2 x-6 z+6 x-2 z=0$
$\Rightarrow 4 x-8 z=0$
$\Rightarrow x-2 z=0$
Thus, the required equation is $x-2 z=0$.
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