Question

Find an equation of a sphere if one of its diameters has endpoints (5,1,6) and (7,3,8)

a. (x-6)² + (y-2)² + (z-7)² = 12
b. (x-5)² + (y-1)² + (z-6)² = 12
c. (x-6)² + (y-2)² + (z-7)² = 144
d. (x-5)² + (y-1)² + (z-6)² = 144

Answer 1

The equation of the sphere with a diameter having endpoints (5,1,6) and (7,3,8) is (x - 6)^2 + (y - 2)^2 + (z - 7)^2 = 12, which corresponds to option (a).

Step-by-step explanation:

To find the equation of a sphere given the endpoints of one of its diameters, use the midpoint formula to find the center of the sphere, and the distance formula to find the radius. Let's apply this procedure to the endpoints given, which are (5,1,6) and (7,3,8).

First, find the center of the sphere by taking the midpoint of the line segment defined by the endpoints. This will be:

• x-coordinate of the center: (5 + 7) / 2 = 6
• y-coordinate of the center: (1 + 3) / 2 = 2
• z-coordinate of the center: (6 + 8) / 2 = 7

So the center of the sphere is at (6, 2, 7).

Next, calculate the radius of the sphere by finding the distance between the center and either endpoint. Use the distance formula:

r = √[(7 - 5)^2 + (3 - 1)^2 + (8 - 6)^2] = √[4 + 4 + 4] = √[12]

Hence, the radius r is √[12], and the radius squared is 12. We can now write the equation of the sphere in standard form:

(x - 6)^2 + (y - 2)^2 + (z - 7)^2 = 12

Thus, the correct option is (a).