Find the Equation of a Sphere with Given Endpoints and Midpoint

[rcmango]

Homework Statement

Find an equation of the sphere one of whose diameters has (1, 3, -5) and
(3, -1, 3) as its endpoints.

Homework Equations

midpoint.

The Attempt at a Solution

i don't understand how to find the midpoint, even though its the mean of:
(2, 1, 1)

i've also been given the hint to find the measurement of a great circle:
sqrt( 4 + 16 + 64 ) = sqrt(80)


which equals: Thus,
(x-2) ^ 2 + (y-1) ^ 2 + (z-1) ^ 2 = 80

please help me solve this problem, i almost got it!

 

[dextercioby]

I think the center's at (2,1,-1) and from here it follows that the radius is $\sqrt{21}$

 

[eprjenkins]

sqrt( 4 + 16 + 64 ) = sqrt(80)?

Shouldn't it be sqrt( 4 + 16 + 64 ) = sqrt(84)

 

[HallsofIvy]

The midpoint of the line segment from (x₀,y₀,z₀) to (x₁,y₁,z₁) is
$$\left(\frac{x_0+x_1}{2},\frac{y_0+y_1}{2},\frac{z_0+z_1}{2}\right)$$
I thought everyone knew that!
The length of the diameter is $\sqrt{84}= 2\sqrt{21}$.
The length of the radius is half that: $\sqrt{21}$