Question

Write the sphere equation in standard form: x^2 + y^2 + z^2 + 8x − 8y + 4z + 27 = 0 CALCULATE the center(x,y,z) and the radius

Answer 1

Hi,

Explanation:

`x^2+y^2+z^2+8x-8y+4z+27=0\\(x^2+2*4x+16)+(y^2-2*4y+16)+(z^2+2*2z+4)-36+27=0\\\\(x+4)^2+(y-4)^2+(z+2)^2=3^2\\`

center is (-4,4,-2)

radius=3

Answer 2

see explanation

Explanation:

the equation of a sphere in standard form is

(x - h)² + (y - k)² + (z - l)² = r²

where (h, k, l ) are the coordinates of the centre and r is the radius

given

x² + y² + z² + 8x - 8y + 4z + 27 = 0

collect x/y/z terms and subtract 27 from both sides

x² + 8x + y² - 8y + z² + 4z = - 27

using the method of completing the square

add ( half the coefficient of the x/y/z terms)² to both sides

x² + 2(4)x + 16 + y² + 2(- 4)y + 16 + z² + 2(2)z + 4 = - 27 + 16 + 16 + 4

(x + 4)² + (y - 4)² + (z + 2)² = 9 ← in standard form

with centre = (- 4, 4, - 2 ) and r =
`√(9)` = 3