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)

  (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

∠acb = 8°

step-by-step explanation:

∠xba lies on the circle and is half the measure of the intercepted arc ax

hence

measure of arc ax = 2 × 42° = 84°

∠acb is outside the circle and is

∠acb = (arcab - arcax ), hence

∠acb = 0.5 (100 - 84) = 0.5 × 16° = 8°

v = 1/3 a^2h

step-by-step explanation:

the volume is 1/3 a^2h.