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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 4, Problem 15P

**(a)**

To determine

**To show:** If *Q* is any point on the parabola where the normal line at *P* intersects the parabola, then the *y-* coordinate of *Q* is smallest when

Expert Solution

The equation of the parabola is given as

On differentiating with respect to *x*,

Thus, the slope of the tangent is,

So, the slope of the normal is,

At the point

By slope point form, the equation of normal is

Use

One is the *x-*coordinate of *P* say *x-*coordinate of *Q* say

Using the value of

Find the derivative of *a* as follows.

Set

On simplification,

Since *a* is a real number, the value of *a* is

Compute the second derivative of *a* as follows.

Clearly for any value of *a*,

That is, the value of *y* coordinate is minimum when

Hence the proof.

**(b)**

To determine

**To show:** If *Q* any point where normal line *P* intersects the parabola then the line segment PQ has the shortest possible length when

Expert Solution

Use the information deduced in part (a).

If *Q* is given by

Let the distance between these two points is

Differentiate the above equation with respect to *a*.

Equate the above equation to 0.

On simplifying the above equation, the only possible value of *a* is

Compute the second derivative of *a* as follows.

As *P* and *Q*.

That is the segment *PQ* has the shortest possible length when

Hence the proof.