x² + y² + z² - 10x - 4y - 14z + 74 = 0
Step-by-step explanation:
The general equation of a sphere is (x-a)² + (y-b)² + (z-c)² = r²
Where x, y, and z are the coordinates of points on the surface of the sphere.
a, b, and c represents the center of the sphere
r is the radius of the sphere. Note that the radius is always the same for all points on the sphere,
In this equation, the radius r is the largest radius that stays in the octant.
In the given question, (5,2,7) is the center of the sphere.
Therefore, substitute this into the general equation to get:
(x-5)²+(y-2)²+(z-7)² = r² ---------------------------------------------(i)
To find the radius r, we have to look at the distance from the center coordinate to each bounding planes xy-plane, xz-plane, and yz-plane.
The distance from the center to the xy-plane is the center of the z coordinate which is 7. The distance from the center to the xz-plane is the center of the y coordinate which is 2. The distance from the center to the yz-plane is the center of the x coordinate which is 5.
Therefore, to determine the radius contained in the first octant, we need to choose the smallest distance so as not to cross into a second octant. That will also be the largest possible radius for it not to cross into a different octant.
The smallest distance therefore is 2. So we substitute r = 2 into equation (i) above to get:
(x-5)²+(y-2)²+(z-7)² = 2²
(x-5)²+(y-2)²+(z-7)² = 4
Therefore (x-5)²+(y-2)²+(z-7)²-4 = 0 ----------------------------------------(ii)
A monic formula is a formula where the highest power of its single variable has a coefficient of 1.
Therefore, we expand equation (ii) in form of a monic formula to get
x² + y² + z² - 10x - 4y - 14z + 74 = 0
The highest power of x², y², and z² is 1
