Use the following information to find the distance between Point A and Point B.

Question

Use the following information to find the distance between Point A and Point B.

Point A is the center of the sphere x² + 4x + y² + z² − 2z = 0.

Point B lies in the middle of the line segment connecting the points P₀(1, 3, 1) and P₁(−3, 5, 1).

Answer 1

The general equation of a sphere is: 

(x - a)² + (y - b)² + (z - c)² = r²

(a, b, c) represents the center of the sphere, r represents the radius, and x, y, and z are the coordinates of the points on the surface of the sphere.

x² + 4x + y² + z² − 2z = 0

Group all the x’s, y’s and z’s. You’ll have:

(x² + 4x + __) + y² + (z² − 2z + __) = 0 + __ + __

This time we’ll do the completing the square by providing constants on the blanks. If your quadratic polynomial is like this: x² + bx + c, to make it perfect square:

c = (b/2)²

For x² + 4x + __, b=4. Therefore c = (4/2)² = 4.

For z² − 2z + __, b=-2. Therefore c = (-2/2)² = 1.

For y² is already a perfect square so y² = (y- 0)². Since you are adding 4 and 1 on one side of the equation, do same thing on the other.

(x² + 4x + 4) + (y- 0)² + (z² − 2z + 1) = 0 + _4_ + _1_

( x + 2)² + (y- 0)² + (z - 1)² = 5.

Therefore, the center is (-2,0,1) and the radius is √5 

To get the midpoint M(x, y, z) of P₀(1, 3, 1) and P₁(−3, 5, 1), we can use the following formula:

(x,y,z) = ( x₁+x₂ , y₁+y₂ , z₁+z₂ )

2 2 2

(x,y,z) = ( 1-3 , 3+5 , 1+1 )

2 2 2

(x,y,z) = (-1,4,1)

So the distance between the center of the sphere (-2,0,1) and the midpoint of the line (-1,4,1) can be obtain using this distance formula:

_______________________

D = √(x - x₀)² + (y - y₀)² + (z - z₀)²

_______________________

D = √(-2-(-1))² + (0 - 4)² + (1 - 1)²

D= √17