--- title: "Radius of curvature of a² = r²cos2θ" subject: "Mathematics I" chapter: 1 key: "semester_1/mathematics_i:curvature-lemniscate-a2-r2cos2theta" source: https://www.bctengineer.com/q/semester_1/mathematics_i/curvature-lemniscate-a2-r2cos2theta site: BCT Engineering --- # Radius of curvature of a² = r²cos2θ Mathematics I · Chapter 1: Derivatives and their Applications - Asked in: 2072 Chaitra - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/mathematics_i/curvature-lemniscate-a2-r2cos2theta - Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1 ## Question Find the radius of curvature at any point (r,θ) for the curve a² = r²cos2θ. ## Short answer By the pedal method: p = a²/r, dp/dr = –a²/r², so ρ = r dr/dp = r³/a² in magnitude. ## Answer **Note:** the curve a² = r² cos 2θ is r²(cos²θ − sin²θ) = a², i.e. x² − y² = a², a rectangular hyperbola (not a lemniscate). We find ρ by the pedal method ρ = r dr/dp. Take logs: ln a² = 2 ln r + ln cos 2θ. Differentiate with respect to θ: 0 = (2/r)(dr/dθ) − 2 tan 2θ, so r dθ/dr = cot 2θ = tan(π/2 − 2θ) So φ = π/2 − 2θ, and p = r sin φ = r cos 2θ = r · a²/r² = a²/r (since cos 2θ = a²/r²). Differentiate: dp/dr = −a²/r². Therefore ρ = r dr/dp = r · (−r²/a²) = −r³/a². Magnitude: **ρ = r³/a²** (the negative sign only shows the direction of the centre of curvature). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. a^2 = r^2 cos 2 theta is actually: A. a lemniscate B. a rectangular hyperbola x^2 - y^2 = a^2 C. a cardioid D. a circle Answer: B. r^2(cos^2 - sin^2) = a^2. (Key point: Nature of curve) 2. phi = A. pi/2 - 2 theta B. pi - 2 theta C. 2 theta D. pi/2 + 2 theta Answer: A. r d theta/dr = cot 2 theta. (Key point: Angle) 3. p = A. r B. a r C. r^2/a D. a^2/r Answer: D. r cos 2 theta = r a^2/r^2. (Key point: Pedal) 4. rho magnitude is: A. r^3/a^2 B. a^2/r^3 C. r^2/a D. 3r/a Answer: A. r dr/dp = -r^3/a^2. (Key point: Result) ## Close variants set in other exams - [Radius of curvature of the astroid](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Acurvature-astroid-any-point) — Mathematics I, chapter 1, asked 2× ## Related in Mathematics I - [Radius of curvature of r^m = a^m cos mθ](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Acurvature-polar-rm-cos-mtheta) — Mathematics I, chapter 1, asked 2× - [Radius of curvature of x³+y³=3axy at origin](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Acurvature-origin-folium) — Mathematics I, chapter 1, asked 3× - [Radius of curvature of r = a(1−cosθ)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Acurvature-cardioid-1-minus-cos) — Mathematics I, chapter 1 - [Pedal equation of r² = a²cos2θ](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Apedal-equation-lemniscate) — Mathematics I, chapter 1, asked 3× - [Curvature where tangent is parallel to x-axis](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Acurvature-y-x2-x-minus-3) — Mathematics I, chapter 1 - [Ellipse curvature at end of major axis](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Acurvature-ellipse-extremity-major-axis) — Mathematics I, chapter 1 ## Related in other subjects - [Vertical curve radius 5000 m, BVC and highest point](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-73bh-vc-5000) — Transportation Engineering I, chapter 3 - [Dependent motion; radius of curvature of particle path](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Adependent-motion-particle-ax-ay) — Applied Mechanics, chapter 8 - [Normal, tangential acceleration; radius of curvature](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aposition-vector-normal-tangential-radius-curvature) — Applied Mechanics, chapter 8 - [Degree of curvature and radius relation](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Adegree-of-curvature-radius-relation) — Surveying II, chapter 6