Q. 91 - Precalculus Enhanced with Graphing Utilities

Question

Descartes’s Method of Equal Roots: Descartes’s 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 (2,4). 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 $(2,4)$ 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

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

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

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

To obtain a unique solution for x, the two roots must be equal; in other words, the discriminant $m^2-4(2m-4)$ must be 0.

Complete the work to get m, and write an equation of the tangent line.

Step-by-Step Solution

Answer

The equation of the tangent line is $y=4x-4$.

Step 1. Given information

Using the quadratic equation given, we can find the value of x. But the tangent at a given point should be unique. In order for the value of x to be unique, the discriminant of the quadratic equation must be zero.

The discriminant of the quadratic equation is $m^2-4(2m-4)$.

$$\begin{gathered} m^2-4(2m-4)=0 \\ {m}^2-8m+16=0 \\ {(m-4)}^2=0m-4=0 \\ m=4 \end{gathered}$$

Step 2. Find the equation for the tangent line.

Substitute $m=4$ in $y=mx+b$;

where $b=4-2m$

$$\begin{gathered} b=4-2\left(4\right) \\ =4-8 \\ =-4 \end{gathered}$$

which gives

$y=4x-4$

which is the equation for the tangent line of $y=x^2$ at $(2,4)$.