Find an equation of the largest sphere with center (5,2,7)(5,2,7) and is contained in the first octant. be sure that your formula is monic.

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

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