Chapter 9, Problem 23CR

To determine

To find: the equation of the tangent line to the parabola.

Answer to Problem 23CR

The equation of the tangent line to the parabola is $y=\frac{1}{4}x-2$ .

Explanation of Solution

Given information:

The equation of the parabola is $y^2=-2x$ and the point is $\left(-8,-4\right)$ .

Calculation: to find the equation of the tangent line, first differentiate the equation of parabola,

  $\begin{array}{l}{y}^2=-2x\\ 2y\frac{dy}{dx}=-2\text{ [differentiate]}\end{array}$

Thus, the slope of tangent line at the point $\left(-8,-4\right)$ is,

  $\begin{array}{l}2y\frac{dy}{dx}=-2\\ 2\left(-4\right)m=-2\text{ [substitute]}\\ m=\frac{1}{4}\text{ [simplify]}\end{array}$

The equation of tangent line of parabola is $y=\frac{1}{4}x+c$ .

Now, substitute the point in the equation,

  $\begin{array}{l}y=\frac{1}{4}x+c\\ -4=\frac{1}{4}\left(-8\right)+c\text{ [substitute]}\\ c=-2\text{ [simplify]}\end{array}$

Thus, the equation of tangent line is $y=\frac{1}{4}x-2$ .