Question

Find the equation of the sphere with points P such that the distance from P to A is twice the distance from P to B. A(-2, 6, 4), B(6, 3, -2)

Answer 1

Answer:

  (x -26/3)² +(y -2)² +(z +4)² = 436/9

Step-by-step explanation:

For P = (x, y, z), the distance to point A is ...

  dA = √((x+2)² +(y-6)² +(z-4)²)

and the distance to point B is ...

  dB = √((x-6)² +(y-3)² +(z+2)²)

We want the former to be twice the latter, so ...

  dA = 2dB

  √((x+2)² +(y-6)² +(z-4)²) = 2√((x-6)² +(y-3)² +(z+2)²)

Squaring both sides gives ...

  (x+2)² +(y-6)² +(z-4)² = 4((x-6)² +(y-3)² +(z+2)²)

Subtracting the left side, we are left with ...

  4(x-6)²-(x+2)² +4(y-3)² -(y-6)² +4(z+2)² -(z-4)² = 0

Expanding gives us the general form equation of the sphere:

  3x² -52x +3y² -12y +3z² +24z +140 = 0

We can divide by 3 and complete the squares to get the standard-form equation:

  (x -26/3)² +(y -2)² +(z +4)² = 436/9