Find that the given equation represents a sphere, and find its center and radius.?

Question

(x^2)+2x+(y^2)+8y+(z^2)-4z=28

Answer 1

Hi,

Find that the given equation represents a sphere, and find its center and radius.?

Complete the squares and add the same amount to the other side:

x² + 2x+ 1 + y² + 8y + 16 + z² - 4z + 4=28 +1 + 16 + 4

Factor each trinomial and combine like terms on the right:

(x + 1)² + (y + 4)² + (z - 2)² = 49

The center is the point (-1,-4,2) and the radius is 7.

I hope that helps!! :-)

Answer 2

You have to complete the square for each of the independent variables.

(x^2+2x)+(y^2+8y)+(z^2-4z)=28
(x+1)^2 - 1 + (y+4)^2 - 16 + (z-2)^2 - 4 = 28
(x+1)^2 + (y+4)^2 + (z-2)^2 = 49 = 7^2

Center is (-1, -4, 2) and radius 7.

Answer 3

You add to both sides as such:
x^2+2x+(2/2)^2+y^2+8x+ (8/2)^2+z^2-4z+(4/2)^2= 28+(2/1)^2+(8/2)^2.
Or, (x^2+2x+1)+(y^2+8y+16)+ (z^2-4z+4)=28+1+16+4=49,
or, (x+1)^2+(y+4)^2+(z-2)^2=49.

So, the center is at (-1,-4,2), and radius is sqr49=7.

Answer 4

(x^2)+2x+(y^2)+8y+(z^2)-4z=28
(x^2+2x)+(y^2+8y)+(z^2-4z)=(1+16+4)-(1+16+4)+28
(x^2+2x+1)+(y^2+8y+16)+(z^2-4z+4)=(1+16+4)+28
(x+1)^2+(y+4)^2+(z-2)^2=49=7^2
which is a function of a sphere
where center of the sphere is (-1,-4,+2)
and radius is 7

Answer 5

The equation of a sphere is
(x-h)² + (y-k)² + (z-w)² = r² for a sphere with center at (h,k,w) and radius r. This one is trivially a sphere of radius 7 located at coordinates
(-1, -4, +2)

Doug

Answer 6

do your own homework!