Find the equation of a sphere if one of its diameters has endpoints: (-6, 4, 0) and (0, 0, 6)

Solution:

If the endpoints of the diameters of a sphere are (-6, 4, 0) and (0, 0, 6) then the diameter of the sphere is:

Diameter = √[(x₂ - x₁ )² + (y₂ - y₁ )² + (z₂ - z₁)²]

(x₁, y₁, z₁) = (-6, 4, 0)

(x₂, y₂, z₂) = (0, 0,6)

The length of the diameter is = √[(0 - (-6) )² + (0 - 4 )² + (6 - 0)²] = √36 + 16 + 36 = √88

The radius of the sphere is = (√88)/2 =(√4 × 22)/2 = (√4 × √22)/2 = (2 × √22)/2 = √22

The center of the sphere will be the midpoint of the diameter and the midpoint will be:

Center of the sphere =[ (x₁ + x₂)/2, (y₁ + y₂)/2 , (z₁ + z₂)/2] = [(-6 + 0)/2, (4 + 0)/2, (0 + 6)/2] = (-3, 2, 3)

The standard form of the equation of the sphere is :

(x - x₀)² + (y - y₀)² + (z - z₀)² = a²

where a is the radius of the sphere

x₀ = -3; y₀ = 2; z₀ = 3 [The coordinates of the center of the sphere]

Therefore the standard equation of the sphere is :

(x - (-3))² + (y - 2)² + (z - 3)² = (√22)²

(x + 3)² + (y - 2)² + (z - 3)² = 22

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Find the equation of a sphere if one of its diameters has endpoints: (-6, 4, 0) and (0, 0, 6)

Summary:

The equation of a sphere if one of its diameters has endpoints: (-6, 4, 0) and (0, 0, 6) is (x + 3)² + (y - 2)² + (z - 3)² = 22