Descartes’ Method of Equal Roots Descartes' method for finding tangents depends on the idea that, for many graphs, the tangent line at a given point is the unique line that intersects the graph at that point only. We apply his method to find an equation of the tangent line to the parabola at the point . See the figure.
First, we know that the equation of the tangent line must be in the form . Using the fact that the point is on the line, we can solve for in terms of and get the equation . Now we want to be the unique solution to the system.
From this system, we get . By using the quadratic formula, we get
To obtain a unique solution for , the two roots must be equal; in other words, the discriminant must be 0. Complete the work to get , and write an equation of the tangent line.
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College Algebra (10th Edition)
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCollege AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning