The 3-D Coordinate System — Question 3

Question 3

Consider the fixed points $$A=(-2,0,0),\qquad B=(4,0,0),$$ and let $P=(x,y,z)$ satisfy the distance condition $$d(P,A)=2d(P,B).$$

Tasks

1. Derive a Cartesian equation for the locus of all possible points $P$.

2. Identify the surface by stating its center and radius.

3. Determine which of $A$ and $B$ lies inside the surface.

4. Explain geometrically what the factor $2$ requires of points on the locus.

Question 3 – Solution

Strategy Square the distance relation, expand, and complete the square. The resulting surface is an Apollonius sphere.

Derivation The condition is $$(x+2)^2+y^2+z^2=4\bigl((x-4)^2+y^2+z^2\bigr).$$ After collecting terms and dividing by $3$, $$x^2-12x+y^2+z^2+20=0.$$ Completing the square gives $$\boxed{(x-6)^2+y^2+z^2=16}.$$ Thus the locus is the sphere centered at $(6,0,0)$ with radius $4$.

Interpretation Point $B$ is $2$ units from the center and lies inside; $A$ is $8$ units from the center and lies outside. A point must remain closer to $B$ because its distance to $A$ is required to be twice as large.

Verification At the axial endpoints $(2,0,0)$ and $(10,0,0)$, the distance pairs are $(4,2)$ and $(12,6)$, respectively, confirming the ratio.