question HELP summer school

Question

find the center and radius of the sphere. 4x^2 + 4y^2 + 4z^2 - 4x + 24z = 10

Answer 1

Given equation of sphere is 4x^2 + 4y^2 + 4z^2 - 4x + 24z = 10

The general form of equation of sphre is x^2 + y^2 +  z^2 +2gx + 2fy + 2hz +c = 0

Divide the given equation with 4

(4x^2 + 4y^2 +4z^2 -4x +24z -10)/4 = 0

Therefore the obtained equation is (x^2 + y^2 + z^2 - 4 + 6z - (10/4) = 0

Compare the obtained eqution with equation of sphere

2g = (-4) , 2f = (0) , 2h = (6) , c = (5/2)

Generally centre (-g , -f  , -h)

Therefore centre is (-(-4/2) , (-0/2) , (-6/2))

Centre = (2 , 0 , -3)

General form of radius is √(g^2 + f ^2 + h^2 -c)

Therefore radius = √((2)^2 + 0^2 + (-3)^2 - (5/2)

=√4 + 9 + -(5/2)

Take L.C.M √(8 + 18 - 5)/2

=√(26 - 5)/2

=√(21)/2

Therefore the centre and the radius of the given sphere is

centre = (2 , 0 , 3) and the radius is √(21/2)