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Calculus IIChapter 3: Parametric Equations and Polar CoordinatesSection 3-6

Polar Coordinates — Question 8

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Question 8

Problem

Which description better reveals the geometry of x2+y2=5xx^2+y^2=5x: Cartesian or polar? Convert it and explain.

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Original worksheet page 1: question and worked solution for 3-6-008
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Question 8 – Solution

See the diagram in the original worksheet below.

Solution

  1. Use the polar–Cartesian relationships x=rcos⁡θ,y=rsin⁡θ,r2=x2+y2.x=r\cos\theta,\qquad y=r\sin\theta,\qquad r^2=x^2+y^2. Equivalent polar coordinates satisfy (r,θ)=(r,θ+2kπ)=(−r,θ+(2k+1)π).(r,\theta)=(r,\theta+2k\pi)=(-r,\theta+(2k+1)\pi).

  2. Apply the identity that matches the requested conversion, symmetry test, or intersection, and then check the resulting point or curve in the original polar equation.

  3. Polar form is r=5cos⁡θr=5\cos\theta, immediately revealing a circle through the pole with diameter 55 along the polar axis.

Original worksheet page 2: question and worked solution for 3-6-008

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