Question

Answer the following questions about the sphere whose equation is given by x2+y2+z2−8x+4y=−4. 1. Find the radius of the sphere. Radius: ????= 2. Find the center of the sphere. Write the center as a point (????,????,c) where ????, ????, and c are numbers. Center:

Answer 1

we have (a,b,c)=(4,-2,0) and R=4 (radius)

Explanation:

since

x²+y²+z²−8x+4y=−4

we have to complete the squares to finish with a equation of the form

(x-a)²+(y-b)²+(z-c)²=R²

that is the equation of a sphere of radius R and centre in (a,b,c)

thus

x²+y²+z²−8x+4y=−4

x²+y²+z²−8x+4y +4 = 0

x²+y²+z²−8x+4y +4 +16-16 =0

(x²−8x + 16) + (y² + 4y + 4 ) + (z²) -16 = 0

(x-4)² + (y+2)² + z² = 16

(x-4)² + (y-(-2))² + (z-0)² = 4²

thus we have a=4 , b= -2 , c= 0 and R=4