I can not seem to get this right
Find an equation of a sphere if one of its diameters has endpoints
(5, 5, 5) and (7, 7, 7).
The center is at (6,6,6)
The diameter has length √12=2√3
so, the sphere is
(x-6)^2 + (y-6)^2 + (z-6)^2 = 3
It'd be better in the future if you provide some of your work when you get stuck.
To find the equation of a sphere, you need to know the center of the sphere and its radius. In this case, you are given two endpoints of a diameter.
Step 1: Find the center of the sphere:
Since the diameter is defined by the endpoints (5, 5, 5) and (7, 7, 7), you can find the center of the sphere by taking the average of the coordinates.
Center = ((5+7)/2, (5+7)/2, (5+7)/2) = (6, 6, 6)
Step 2: Find the radius of the sphere:
The radius of the sphere is half the length of the diameter. You can use the distance formula to find the length of the diameter, and then divide it by 2 to get the radius.
Diameter = √((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)
Diameter = √((7-5)^2 + (7-5)^2 + (7-5)^2)
Diameter = √(2^2 + 2^2 + 2^2)
Diameter = √(4 + 4 + 4)
Diameter = √12 = 2√3
Radius = Diameter/2 = (2√3)/2 = √3
Step 3: Write the equation of the sphere:
The equation of a sphere with center (h, k, l) and radius r is given by:
(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2
Substituting the values we obtained earlier, the equation of the sphere is:
(x-6)^2 + (y-6)^2 + (z-6)^2 = (√3)^2
Simplifying the equation gives us:
(x-6)^2 + (y-6)^2 + (z-6)^2 = 3