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 there's no need to adjust $z^2$ since its coefficient is already perfect. So, we get:$$x^2 - 4x + 4 + y^2 - 8y + 16 + z^2 = 16$$$$ (x - 2)^2 + (y - 4)^2 + z^2 = 16 $$This shows the center of the sphere is at $C(2, 4, 0)$ and the radius $r = \sqrt{16} = 4$. None of the provided options match this result.

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 $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 4 - 8 y = 0$