Question

Write the equation of the sphere in standard form. x2 + y2 + z2 + 8x − 6y + 2z + 17 = 0

Answer 1

The equation of the sphere in standard form is (x + 4)^2 + (y - 3)^2 + (z + 1)^2 = 9, with a center at (-4, 3, -1) and a radius of 3 units.

Step-by-step explanation:

To write the given equation of a sphere in standard form, we will complete the square for the variables x, y, and z. The standard form of a sphere's equation is (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2, where (h, k, l) is the center of the sphere, and r is the radius. The given equation is x^2 + y^2 + z^2 + 8x − 6y + 2z + 17 = 0. To complete the square, we group the x, y, and z terms together:

x^2 + 8x + y^2 - 6y + z^2 + 2z = -17

We then add and subtract the necessary constants to complete the square:

(x^2 + 8x + 16) + (y^2 - 6y + 9) + (z^2 + 2z + 1) = -17 + 16 + 9 + 1

This simplifies to:

(x + 4)^2 + (y - 3)^2 + (z + 1)^2 = 9

The center of the sphere is (-4, 3, -1), and the radius is 3 units.

Answer 2

The given equation of sphere is:

`x^(2)+y^(2)+8x-6y+2z+17=0`

Regrouping the variables together,

`(x^(2)+8x)+(y^(2)-6y)+(z^(2)+2z)+17=0`

We will use completing the square method, in this method we will consider the coefficients of x, y and z individually, then we will divide the coefficients by 2. Further, we will square the number obtained after dividing. At last we will add and subtract the number obtained.

`(x^(2)+8x+16)-16+(y^(2)-6y+9)-9+(z^(2)+2z+1)-1+17=0`

`(x+4)^(2)+(y-3)^(2)+(z+1)^(2)=26-17`

`(x+4)^(2)+(y-3)^(2)+(z+1)^(2)=9`

`(x+4)^(2)+(y-3)^(2)+(z+1)^(2)=(3)^(2)`

is the standard form of the sphere with center
`(-4,3,-1)` and radius as 3.