I took a quiz in my vector calc class. I got a few points off, and am trying to figure out what I did wrong. Here's the question:
Show that the equation represents a sphere, and find its center and radius: $3x^2 - 6x + 3y^2 - 3y + 3z^2 = 9$
For the radius, I got: $R = \sqrt{5/4} = \sqrt{(x-1)^2 + (y - 1/2)^2 + z^2}$
And for the center I got: $C=(1, 1/2, 0)$
Any guidance would be much appreciated.
I'll work through it and see what I get.
$\begin{array}\\ 3x^2 - 6x + 3y^2 - 3y + 3z^2 = 9 \iff &x^2 - 2x + y^2 - y + z^2 = 3\\ \iff &x^2 - 2x+1 + y^2 - y+\frac14 + z^2 = 3+1+\frac14 = 17/4\\ &\qquad\text{You seem to have lost the 3 here}\\ \iff &(x-1)^2+(y-1/2)^2 + z^2= 17/4\\ \end{array} $
So center is at $(1, 1/2, 0)$ and the radius is $\sqrt{17}/2$.