Find an equation of the sphere that passes through the point (7, 5, -3) and has a center (5, 8, 1).

Solution:

We have equation of sphere centre at (h, k, l) and radius r as (x - h)² + (y - k)² + (z - l)² = r²

Given centre = (h, k, l) = (5, 8, 1) and the sphere passes through a point (7, 5, -3).

Radius = distance between the centre (5, 8, 1) and the point (7, 5, -3).

∴ r = √[(5 - 7)² + (8 - 5)² + (1 - (-3))²]

r = √29

Equation of sphere is (x - h)² + (y - k)² + (z - l)² = r²

⇒ (x - 5)² + (y - 8)² + (z - 1)² = (√29)²

⇒ x² + y² + z² -10x -16y - 2z + 25 + 64 + 1 = 29

⇒ x² + y² + z² - 10x - 16y - 2z + 61

Find an equation of the sphere that passes through the point (7, 5, -3) and has a center (5, 8, 1).

Summary:

The equation of the sphere that passes through the point (7, 5, -3) and has a center at (5, 8, 1), is x² + y² + z² - 10x - 16y - 2z + 61.