What happens to a line in polar coordinates when origin is moved and rotated in cartesian coordinates?

Let's say we have an Archimedean spiral in Cartesian coordinates. This corresponds to a line in polar system (i.e. $r=a\theta+b$).

Now if I move the origin of the Cartesian coordinates system to $\begin{bmatrix} x_1 \\ y_1 \end{bmatrix}$ and rotate its axes by $\alpha$ counter-clockwise, the spiral is still a spiral in Cartesian system (the shape does not change).

Therefore, it should still be a line in polar system too (i.e. $r_1 = a_1\theta_1+b_1$).

So what is the equation for the second spiral in polar system (in terms of $x_1$ and $y_1$ and $\alpha$)?

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Your reasoning is not quite right. An Archimedean spiral will only have a nice simple equation of the form $r = a\theta + b$ if you use the center of the spiral as your origin. If you use some other origin, the equation will be a mess.
The same is true of most (maybe all?) types of curves. Circles are the simplest example. A circle of radius $k$ has the polar equation $r=k$ if you place the coordinate origin at its center. If you place the origin somewhere else, you'll get a much more complex equation.