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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.1, Problem 73E

To determine

**To find:** The number of lines that are normal to the parabola

Expert Solution

Three normal lines to the parabola that passes through the point

One normal line to the parabola that passes through the point

**Derivative rule:**

Power rule:

**Calculation:**

The derivative of the curve

Apply the power rule (1) and simplify the expression,

Thus, the derivative of the curve is

Therefore, the slope of the tangent line to the curve is 2*x*.

Obtain the slope of the normal line to the curve by using the slope of the tangent line.

Since every point of the parabola is of the form *a*.

Here, the tangent line is perpendicular to the normal line. That is, if

This implies that, the slope of the normal line to the curve is

Note that, the slope of the line passing through the points

Here, the normal line passing through the points

The slope of the normal line is computed as follows.

Since the slope of normal line is

The above equation has two solution if

Since the *y*-axis is normal to the parabola and passes through the point *c*).

Therefore, there are three normal lines to the parabola that passes through the point