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Show an equivalence between given sets in Cartesian and polar coordinates

by RoRi

Show that the sets of points satisfying

    \[ x^2 + y^2 - x = \sqrt{x^2 + y^2}; \qquad r = 1 + \cos \theta. \]

are equal.

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

    \begin{align*} x^2 + y^2 - x = \sqrt{x^2 + y^2} && \implies && r^2 \cos^2 \theta + r^2 \sin^2 \theta - r \cos \theta &= \sqrt{r^2 \cos^2 \theta + r^2 \sin^2 \theta} \\ && \implies && r^2 - r \cos \theta &= r \\ && \implies && 1 + \cos \theta &= r. \qquad \blacksquare \end{align*}

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