Question

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

• A. $x-2z=0$
• B. $2x-z=0$
• C. $2x+y=0$
• D. $x-2y=0$

Answer 1

The correct option is A $x-2z=0$

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

then $PA=PB$

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

Squaring both sides

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

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

$-2x-6z+10=-6x+2z+10$

$4x-8z=0$

$x-2z=0$