Find the equation of a sphere if one of its diameters has endpoints: (7, 6, -3) and (11, 10, 1).

Find the equation of a sphere if one of its diameters has endpoints: (7, 6, -3) and (11, 10, 1).

The center is the midpoint

((7+11)/2, (6+10)/2, (-3+1)/2) = (9, 8, -1).

The radius is half the distance between the points

r = ½√((7-11)² + (6 - 10)² + (-3-1)²) = 2√(3).

(x - 9)² + (y - 8)² + (z + 1)² = 12.

Then the center is at the avg of the corresponding coordinates

(9, 8, -1)

Then find the length of the diameter and divide by 2 to get r

r =
sqrt[(11-7)^2+(10-6)^2+(1-7)^2]/2 =
sqrt[(4)^2+(4)^2+(-6)^2]/2 =
sqrt[16+16+36]/2 =
sqrt[68]/2 =
2*sqrt[17]/2 =
sqrt[17]

Then the equation is

(x-a)^2 + (y-b)^2 + (z-c)^2 =r^2 Where the sphere is centered at (a,b,c) with a radius r.

(x-9)^2 + (y-8)^2 + (z+1)^2 = 17

I hope this helps