Find the center and the radius of the sphere that has the given equation. $$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 8 y = 0$$
A) $( 2 , - 4,0 ) , 2 \sqrt { 5 }$
B) $( - 2,4,0 ) , 2 \sqrt { 5 }$
C) $( - 2,4,0 ) , 20$
D) $( 2 , - 4,0 ) , 20$

Question

Quizplus · Accepted Answer

To find the center and radius of the sphere given by the equation $x^2 + y^2 + z^2 - 4x + 8y = 0$, we complete the square for $x$ and $y$. The equation can be rewritten as:$$x^2 - 4x + y^2 + 8y + z^2 = 0$$Adding and subtracting the necessary terms to complete the square:$$(x^2 - 4x + 4) + (y^2 + 8y + 16) + z^2 = 4 + 16$$$$(x - 2)^2 + (y + 4)^2 + z^2 = 20$$This shows the center of the sphere is at $(2, -4, 0)$ and the radius is $\sqrt{20}$ or $2\sqrt{5}$.

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

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