Given the helix curve: $$r(t) = 5\sin(t) \,\mathbf i + 12t\,\mathbf j - 5\cos(t)\,\mathbf k$$
What is the equation of the circular cylindrical surface which contains the helix curve and is parallel to the y-axis?
I think the answer to this problem is $x^2 + z^2 = 25$ (I'm not 100% sure). If so, why is that the answer? If I'm wrong, what is the correct solution?
You're right.
A line parallel to the $Y$ axis is given by a pair of $x$ and $z$ coordinates; every point on such line has the same two values of $x$ and $z$ and they differ in $y$ only.
Similary, a cylindrical surface parallel to the $Y$ axis, being a union of lines parallel to the $Y$ axis, is given by an $X$-$Z$ equation; every point of the surface has the same two values of $x$ and $z$ coordinates as some point satisfying the equation.
To find the$X$-$Z$ equation you just need to drop $y$ coordinate from $r(t)$. Every point $r$ on the helix is then projected parallel to $Y$ to the $XZ$ plane: $$r_{XZ}(t) = 5\sin(t)\,\mathbf i - 5\cos(t)\,\mathbf k$$ that is: $$r_{XZ}(t) = (5\sin t, -5\cos t)$$ or $$\begin{cases}x_r(t) = 5\sin t \\ z_r(t) = -5\cos t\end{cases}$$ If we square both sides of both equations we get $$\begin{cases}x^2 = 25\sin^2 t \\ z^2 = 25\cos^2 t\end{cases}$$ which, by the most known trigonometric identity, becomes $$x^2+z^2 = 25(\sin^2 t + \cos^2 t) = 25$$