In Problems 9096, use Descartes method from Problem 89 to find an equation of the tangent line to each graph
In Problems 90–96, use Descartes’ method from Problem 89 to find an equation of the tangent line to each graph at the given point.
x2 − y2 = 3; at (2, 1)
Data from problem 89
Descartes’ method for finding tangent lines 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 use his method to find an equation of the tangent line to the parabola y = x2 at the point (2, 4). See the figure.
First, an equation of the tangent line can be written as y = mx + b. Using the fact that the point (2, 4) is on the line, we can solve for b in terms of m and get the equation y = mx + (4 − 2m). Now we want (2, 4) to be the unique solution to the system
From this system, we get x2 − mx + (2m − 4) = 0. Using the quadratic formula, we get
To obtain a unique solution for x, the two roots must be equal; in other words, the discriminant m2 − 4(2m − 4) must be 0. Complete the work to get m, and write an equation of the tangent line.
This problem has been solved!
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