Reflecting on parabolas
David E. Dobbs
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA.
Pages 45-77 | Received 15 July 2023, Accepted 08 October 2023, Published 30 June 2024
Abstract
Two known results are proven in this teaching note. First, a proof of the reflection property of parabolas is given; that proof would be accessible early in a calculus class or in a course that combines precalculus with an introduction to differential calculus. Second, that reflection property plays a key role in a characterization of parabolas; that proof solves an initial value problem concerning a first order ordinary differential equation, and so it would be accessible early in a course on differential equations or in some courses on integral calculus. A closing remark discusses characterization results and classification results.
Keywords: Euclidean analytic geometry, parabola, focus, directrix, tangential half-line, vector, dot product, inverse cosine function, derivative, initial value problem.
MSC numbers: Primary 51-02; Secondary 51N20, 33B10, 97G70, 26A06, 34-01.
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