Question

Find the equation of the set of the points P such that its distances from the points A(3, 4, -5) and B(-2, 1, 4) are equal.

Answer 1

Let the point be P(x, y, z)

Given

A= (3, 4, -5)

B= (-2, 1, 4)

AP = BP $\Rightarrow A{P}^2=B{P}^2$

$A{P}^2=(x-3{)}^2+(y-4{)}^2+(z+5{)}^2$

$B{P}^2=(x+2{)}^2+(y-1{)}^2+(z-4{)}^2$

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

$\Rightarrow (x-3{)}^2+(y-4{)}^2+(z+5{)}^2$

=$(x+2{)}^2+(y-1{)}^2+(z-4{)}^2$

All square terms will be cancelled on both sides, we get

$-6x+9-8y+16+10z+25$

$=4x+4-2y+1-8z+16$

$10x+6y-18z-29=0$ is the required equation.