Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 4 - 8 y = 0$
A) $C ( 2,2,4 ) ; r = \sqrt { 2 }$
B) $C ( 0,2,4 ) ; r = 4$
C) $C ( 2,4,0 ) ; r = \sqrt { 2 }$
D) $C ( 2,4,0 ) ; r= 2$
E)none of these

Question

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 4 - 8 y = 0$
A) $C ( 2,2,4 ) ; r = \sqrt { 2 }$ 
B) $C ( 0,2,4 ) ; r = 4$ 
C) $C ( 2,4,0 ) ; r = \sqrt { 2 }$ 
D) $C ( 2,4,0 ) ; r= 2$ 
E)none of these

Quizplus · Accepted Answer

To find the center and radius of the sphere given by the equation $x^2 + y^2 + z^2 - 4x + 4 - 8y = 0$, we complete the squares for $x$, $y$, and $z$. The equation can be rewritten as:$$x^2 - 4x + y^2 - 8y + z^2 + 4 = 0$$Completing the square for $x$ and $y$, we add and subtract $(4/2)^2 = 4$ for $x$ and $(8/2)^2 = 16$ for $y$, and adjust the constant term accordingly:$$(x^2 - 4x + 4) + (y^2 - 8y + 16) + z^2 = 4 + 16 - 4$$$$(x - 2)^2 + (y - 4)^2 + z^2 = 16$$This represents a sphere with center $C(2, 4, 0)$ and radius $r = \sqrt{16} = 4$. Since none of the provided options correctly matches both the center and the radius, the correct answer is E) none of these.

Find the Center and Radius of the Sphere x2+y2+z2−4x+4−8y=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 4 - 8 y = 0x2+y2+z2−4x+4−8y=0

Find the Center and Radius of the Sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 4 - 8 y = 0$