# 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} = r^{2}

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} = r^{2}

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

⇒ x^{2} + y^{2} + z^{2} -10x -16y - 2z + 25 + 64 + 1 = 29

⇒ x^{2} + y^{2} + z^{2} - 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 x^{2} + y^{2} + z^{2} - 10x - 16y - 2z + 61.

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