To find the equation of a sphere, we use the standard form:
(x – h)2 + (y – k)2 + (z – l)2 = r2
Here, (h, k, l) is the center of the sphere and r is the radius. Based on the information provided, the center of the sphere is (3, 8, 1).
Next, we need to calculate the radius, which is the distance from the center of the sphere to any point on its surface. The point (4, 3, 1) lies on the surface of the sphere. We can find the radius using the distance formula:
Distance (r) = √((x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2)
Substituting our values:
(x1, y1, z1) = (3, 8, 1) (center)
(x2, y2, z2) = (4, 3, 1) (point on the sphere)
So we calculate:
r = √((4 – 3)2 + (3 – 8)2 + (1 – 1)2)
= √(1 + 25 + 0)
= √26
With this radius, we can now plug the center and the radius into the sphere’s equation:
(x – 3)2 + (y – 8)2 + (z – 1)2 = (√26)2
This simplifies to:
(x – 3)2 + (y – 8)2 + (z – 1)2 = 26
Thus, the equation of the sphere is:
(x – 3)2 + (y – 8)2 + (z – 1)2 = 26