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Show an equivalence between given sets in Cartesian and polar coordinates

Show that the sets of points satisfying

    \[ (x-1)^2 + y^2 = 1 \qquad \text{and} \qquad r = 2 \cos \theta, \quad \cos \theta > 0 \]

are equal.


Letting x = r \cos \theta and y = r \sin \theta we plug in to the given Cartesian equation,

    \begin{align*}  (x-1)^2 + y^2 = 1 && \implies && (r \cos \theta - 1)^2 + r^2 \sin^2 \theta &= 1 \\  && \implies && r^2 \cos^2 \theta - 2 r \cos \theta + 1 + r^2 \sin^2 \theta &=1 \\  && \implies && r^2 (\cos^2 \theta + \sin^2 \theta) - 2 r \cos \theta &= 0 \\  && \implies && r &= 2 \cos \theta, \end{align*}

where \cos \theta > 0 so r > 0 (hence, we can divide by r) in the second to last line.

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