To find the equation of a sphere, we use the standard formula:
(x – h)² + (y – k)² + (z – l)² = r²
In this formula, (h, k, l) represents the center of the sphere, and r is its radius.
Given that the center of the sphere is (3, 8, 1), we can assign:
- h = 3
- k = 8
- l = 1
Next, we need to find the radius, r. The radius can be determined by calculating the distance from the center to the given point (4, 3, 1).
We can use the distance formula:
r = √((x2 – x1)² + (y2 – y1)² + (z2 – z1)²)
Substituting in our center (3, 8, 1) and point (4, 3, 1):
r = √((4 – 3)² + (3 – 8)² + (1 – 1)²)
This simplifies to:
r = √(1² + (-5)² + 0²)
r = √(1 + 25 + 0)
r = √26
Now we have r, so we can substitute back 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 that passes through the point (4, 3, 1) and has its center at (3, 8, 1) is:
(x – 3)² + (y – 8)² + (z – 1)² = 26