# The number of lines that are normal to the parabola y = x 2 such that it passes through the point ( 0 , c ) when c &gt; 1 2 and c ≤ 1 2 . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.1, Problem 73E
To determine

## To find: The number of lines that are normal to the parabola y=x2 such that it passes through the point (0,c) when c>12 and c≤12.

Expert Solution

Three normal lines to the parabola that passes through the point (0,c) if c>12.

One normal line to the parabola that passes through the point (0,c) if c12.

### Explanation of Solution

Derivative rule:

Power rule: ddx(xn)=nxn1

Calculation:

The derivative of the curve y=x2 is computed as follows.

dydx=ddx(x2)

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

dydx=2x21=2x

Thus, the derivative of the curve is dydx=2x.

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

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,a2), the slope of the tangent to the curve at (a,a2) is 2a.

Here, the tangent line is perpendicular to the normal line. That is, if m1 and m2 are the slopes of tangent line and normal line, then m1m2=1.

This implies that, the slope of the normal line to the curve is 12a.

Note that, the slope of the line passing through the points (x1,x2) and (y1,y2) is m=y2y1x2x1.

Here, the normal line passing through the points (0,c) and (a,a2).

The slope of the normal line is computed as follows.

m=a2ca0=a2ca

Since the slope of normal line is 12a, the equation becomes,

12a=a2ca12=a2ca2=c12

The above equation has two solution if c>12, one solution if c=12 and no solution if c<12.

Since the y-axis is normal to the parabola and passes through the point (0,c) (independent form c).

Therefore, there are three normal lines to the parabola that passes through the point (0,c) if c>12 and one normal line to the parabola that passes through the point (0,c) if c12.

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