Polar intersections
Objective. Identify equivalent representations and intersections.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
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Learn: the key idea
Finding where two polar curves intersect requires solving their equations simultaneously, but care is needed because the same point can be represented by different (r, θ) pairs, such as (r, θ) and (r, θ+2π), or (−r, θ+π). Simply equating f(θ) = g(θ) may miss intersections where the curves pass through the same point using different angle representations, especially at the pole r=0. A thorough solution checks both the direct equation and whether both curves pass through the pole at any angle.
Worked example
Find the points of intersection (other than the pole) of the polar curves r = 2 and r = 4 cos θ, for θ in [0, 2π).
- 1Set the two expressions for r equal: 2 = 4 cos θ
- 2Solve: cos θ = 1/2, so θ = π/3 or θ = −π/3 (equivalently 5π/3)
- 3Check both lie in the given range: θ = π/3 and θ = 5π/3 are valid
- 4At both angles, r=2, giving intersection points (2, π/3) and (2, 5π/3)
Answer: (2, π/3) and (2, 5π/3)
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Practice
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Start practisingCommon mistake. Students solve f(θ)=g(θ) only once and miss intersections at the pole, where both curves may reach r=0 at different θ values yet still meet at the same point, the origin. Always check separately whether r=0 is achievable on both curves.
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