# In Problems 9096, use Descartes method from Problem 89 to find an equation of the tangent line to each graph

## Question:

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.

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Question Details
Chapter # 10
Section: Exercise 10.6
Problem: 95
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