Chapter 8.6, Problem 91AE

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 $y=x^2$ at the point $\left(2,4\right)$. See the figure.

First, we know that the equation of the tangent line must be in the form $y=mx+b$. Using the fact that the point $\left(2,4\right)$ is on the line, we can solve for $b$ in terms of $m$ and get the equation $y=mx+\left(4-2m\right)$. Now we want $\left(2,4\right)$ to be the unique solution to the system.

$\{\begin{array}{l}y=x^2\\ y=mx+4-2m\end{array}$

From this system, we get $x^2-mx+\left(2m-4\right)=0$. By using the quadratic formula, we get

$x=\frac{m\pm \sqrt{m^2-4\left(2m-4\right)}}{2}$

To obtain a unique solution for $x$, the two roots must be equal; in other words, the discriminant $m^2-4\left(2m-4\right)$ must be 0. Complete the work to get $m$, and write an equation of the tangent line.