Descartess Method of Equal Roots Descartess method for finding tangents depends on the

Chapter 8, Problem 91

(choose chapter or problem)

Descartess Method of Equal Roots Descartess 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 will 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 b in terms of m and get the equation Now we want to be the unique solution to the systemby = x2y = mx + 4 - 2my = mx + 14 - 2m2. 12, 42y = mx + b. 12, 42From this system, we get By using the quadratic formula, we get To obtain a unique solution for x, the two roots must be equal; in other words, the discriminant must be 0. Complete the work to get m, and write an equation of the tangent line.