Quadric obtained by rotation of a line I have a line of equations:
\begin{align}r:
\begin{cases}
x = 2+2t \\
y = 2 - t \\
z=t
\end{cases}
\end{align}
and another line of equations:
\begin{align}s:
\begin{cases}
x = 3 \\
y = 1 \\
z=k
\end{cases}
\end{align}
I have to determine what quadric I get when i rotate line $r$ about $s$.
Every single point of the line that rotates creates a circumference of equation 
$x^2+y^2=r^2$
hence all those infinte points creates a quadric.
What is the best and fastest way to get to the result?

We have line of equation
\begin{align}r:
\begin{cases}
x=f(t)=2+2t\\
y=g(t)=2-t\\
z=h(t)=t
\end{cases}
\end{align}
and the axis of the rotation
\begin{align}s:
\begin{cases}
x=x_0+at=3\\
y=y_0+bt=1\\
z=z_0+ct=k
\end{cases}
\end{align}
Let's choose a generic point on line $s$ and let it be $C(3,1,0)$.
We have to find the plane through $C$ and perpendiular to $r$.
The plane has generic equations
$a(x-f(t))+b(y-g(t))+c(z-h(t))=0$
So our plane has the same parameters as $s$ $(0,0,1)$ so our plane is
$z-t=0$, in particular $t=z$.
Now we need to find the sphere with centre in $P(3,1,0)$ and radius $\overline{CP}$ knowing that $P$ is a point on line $r$ and has equations $P(f(t), g(t), h(t))$.
The generic sphere has equation:
$(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=(f(t)-x_0)^2+(g(t)-y_0)^2+(h(t)-z_0)^2$
Then our equations becomes
$(x-3)^2+(y-1)^2 +(z-0)^2 = (2+2t-3)^2+(2-t-1)^2+(t-0)^2$
$x^2-6x+9+y^2-2y+1+z^2 = (2t-1)^2+(1-t)^2+t^2$
$x^2-6x+9+y^2-2y+1+z^2 = 4t^2-4t+1+1+t^2-2t+t^2$
$x^2-6x+9+y^2-2y+1+z^2 = 4z^2-4z+1+1+z^2-2z+z^2$
And the equation that we get is
$x^2+y^2-5z^2-6x-2y+6z+8=0$
It is an hyperboloid as we can verifiy with the matrices
\begin{align}I_3=
\begin{vmatrix}
1 & 0 & 0 & -6 \\
0 & 1 & 0 & -2 \\
0 & 0 & -5 & 6 \\
-6 & -2 & 6 & 8
\end{vmatrix} = 124
\end{align}
\begin{align}I_2=
\begin{vmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -5
\end{vmatrix} = -5
\end{align}
We have $det(I_3)>0$ and $det(I_2)\ne0$, $I_2$'s Eigenvalues are $\lambda_1,2 = -1$ and $\lambda_3 = 5$ which indicates that this equation represents an hyperboloid of one sheets

