Normal line to a parabola at a given point: Find the equation of the normal line to

at the point .

We shall work with the fact that the normal line is perpendicular to the tangent line.

First we need to find the slope at any point, i. e. the derivative:

.

At the slope is .

Consequently, the tangent line must have this slope, and the slope of the normal is

. So the equation of the normal line in slope-intercept form is

Since this line goes through the given point, we can use its coordinates in this equation to determine

the y-intercept, b:

, or

We can combine these results to get the equation of the tangent line as