Find the center and radius of the sphere with the equation $x ^ { 2 } + 2 x + y ^ { 2 } - 2 y + z ^ { 2 } - 4 z + 5 = 0$
A) $( - 1,1,2 ) , r = 1$
B) $( 1 , - 1 , - 2 ) , r = 1$
C) $( - 1,1,2 ) , r = 5$
D) $( - 1,1,2 ) , r = \sqrt { 5 }$

Question

Quizplus · Accepted Answer

To find the center and radius of the sphere, we complete the squares for $x$, $y$, and $z$. The equation is given by $x^2 + 2x + y^2 - 2y + z^2 - 4z + 5 = 0$. Completing the square for each variable:- For $x^2 + 2x$, we add and subtract $(2/2)^2 = 1$, resulting in $x^2 + 2x + 1$.- For $y^2 - 2y$, we add and subtract $(-2/2)^2 = 1$, resulting in $y^2 - 2y + 1$.- For $z^2 - 4z$, we add and subtract $(-4/2)^2 = 4$, resulting in $z^2 - 4z + 4$.Adding these adjustments to both sides of the equation, we get:$$x^2 + 2x + 1 + y^2 - 2y + 1 + z^2 - 4z + 4 + 5 = 1 + 1 + 4 + 5$$Simplifying, we have:$$(x + 1)^2 + (y - 1)^2 + (z - 2)^2 = 11$$Subtracting the added constants from both sides:$$(x + 1)^2 + (y - 1)^2 + (z - 2)^2 = 11 - 5$$$$(x + 1)^2 + (y - 1)^2 + (z - 2)^2 = 6$$However, there seems to be a mistake in my calculation of the radius. Correcting this, the equation should actually simplify to:$$(x + 1)^2 + (y - 1)^2 + (z - 2)^2 = 1$$This means the center of the sphere is $(-1, 1, 2)$ and the radius $r$ is the square root of the right side of the equation, which is $1$. Therefore, the correct answer is $( - 1,1,2 ) , r = 1$.

Find the Center and Radius of the Sphere with the Equation