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Show an equivalence between given sets in Cartesian and polar coordinates

by RoRi

Show that the sets of points satisfying

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

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.

3 comments

  1. Kalle says:

    The sets are not equal.

    (x, y) == (0, 0) satisfies the left equation. The origin expressed as polar coordinates is (0, theta), but the equations on the right say that cos(theta)>0. Therefore, r = 2*cos(theta) > 0. Therefore, the Origin is not included in the set defined by the equations on the right.

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