How to find the equation of a sphere given end points of one of it's diameters?

Question

Given the endpoints of a sphere's diameter (2, 1, 4) and (4, 3, 10) find the equation of that sphere

To get the center point, I find the midpoint on each axis, correct? And do I use the distance formula to find the length of the diameter to get the radius?

Answer 1

center point (3, 1.5, 3)

radius;

r' = sqrt(4 + 36) = sqrt(40)

r = sqrt(40 + 4) = sqrt(44)

Answer 2

To get the center point, I find the midpoint on each axis, correct? And do I use the distance formula to find the length of the diameter to get the radius?

Yes, and yes...

center: (3,2,7)

diameter = sqrt[(4-2)^2 + (3-1)^2 + (10-4)^2]

diameter = sqrt(2^2 + 2^2 + 6^2) = sqrt(4 + 4 + 36) =sqrt(44)

radius = sqrt(44) / 2

we can leave it like this, since the RHS of the equation will be r^2, or 44/4 = 11

equation ==> (x - 3)^2 + (y - 2)^2 + (z - 7)^2 = 44/4 = 11

Answer 3

Your questions are both correct.

Center is (3,2,7). Diameter is

sqrt(2^2 + 2^2 + 6^2) = sqrt(44) = 2sqrt11, so the radius is just sqrt11.

Eqn of sphere is

(x-3)^2 + (y-2)^2 + (z-7)^2 = 11

Answer 4

you got it. once you have the radius, just plug it in:

x^2+y^2+z^2=r^2