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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.5, Problem 50E

To determine

**To find:** The point such that the normal line intersect the ellipse at second time and to sketch the equation of the ellipse and the normal line.

Expert Solution

The normal line is intersect to the ellipse second time at the point

**Given:**

The equation of the ellipse is

**Derivative rules:**

*Product rule:*

*Chain rule:*

**Calculation:**

Obtain the slope of the normal line to the ellipse.

Differentiate *x*,

Apply the product rule and the chain rule,

Combine the

Thus, the derivative of the equation is

The slope of the tangent to the ellipse at

Substitute

Thus, the slope of the tangent to the ellipse at *m =* 1.

Note that, if the slope of the tangent line at

Thus, the slope of the normal line to the ellipse at

Obtain the equation of the normal at

It is required that the normal line is intersect to the ellipse at second time.

Substitute

Substitute

Thus, the points

Therefore, the normal line is intersect to the ellipse at second time at

**Graph:**

Use online graphing calculator to sketch the equation of the ellipse and the normal line as shown in Figure 1.

From Figure 1, it is observed that the normal line