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CalculusCalculus10th Edition

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Chapter 10.1, Problem 84E
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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Proof Prove Theorem 10.4 by showing that the tangent line to an ellipse at a point P makes equal angles with lines through P and the foci (see figure). [Hint: (1) Find the slope of the tangent line at P, (2) find the slopes of the lines through P and each focus, and (3) use the formula for the tangent of the angle between two lines.]

To determine

To prove: The theorem of reflective property of an ellipse which states, “If P is a point on an ellipse then the tangent line to the ellipse at point P makes equal angles with the lines through P and the foci.”.

Explanation

Given:

The figure,

And, the theorem, “If P is a point on an ellipse then the tangent line to the ellipse at point P makes equal angles with the lines through P and the foci.”.

Formula used:

The angle,α between the two lines whose slopes are m1 and m2 is given by,

α=tan−1(|m1−m21+m1m2|)

Proof:

Consider the given diagram,

Now, the equation of the ellipse is, x2a2+y2b2=1.

Now the equation, x2a2+y2b2=1 is equivalent to, b2x2+a2y2=a2b2.

Also in an ellipse, a2−b2=c2.

Differentiate the curve, x2a2+y2b2=1 with respect to x.

So,

2xa2+2yy′b2=0y′=−b2xa2y

Thus, the slope of the tangent line at any point on the curve is given by, −b2xa2y.

The slope of the tangent at the point P(x0,y0) is given by, −b2x0a2y0.

Now, the slope of the line passing through (−c,0) and (x0,y0) is given by,

m1=y0−0x0−(−c)=y0x0+c

The slope of the line is, y0x0+c.

Now, use the formula, α=tan−1(|m1−m21+m1m2|) to find the angle between the tangent line and the line passing through P and (−c,0).

So,

β=tan−1(|−b2x0a2y0−y0x0+c1+(−b2x0a2y0)(y0x0+c)|)=tan−1(|−b2x02−b2cx0−a2y02a2y0(x0+c)a2x0y0+a2cy0−b2x0y0a2y0(x0+c)|)=tan−1(|b2x02+b2cx0+a2y02a2x0y0+a2cy0−b2x0y0|)=tan−1(|b2x02+a2y02+b2cx0x0y0(a2−b2)+a2cy0|)

Now, put b2x2+a2y2=a2b2 and a2−b2=c2 in the expression, tan−1(|b2x02+a2y02+b2cx0x0y0(a2−b2)+a2cy0|)

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True or False Label each of the following statements as either true or false. 1. Every cubic equation over the ...

Elements Of Modern Algebra

Find each value requested for the distribution of scores in the following table. X Y 6 1 5 2 4 2 3 4 2 3 1 2 n ...

Statistics for The Behavioral Sciences (MindTap Course List)

For the following set of scores, find the value of each expression: a. X b. (X)2 c. X2 d. (X +3)

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

For Problems 51-78, simplify each of the numerical expressions. Objective 3 (5934)(6927)

Intermediate Algebra

Assume that the distribution of scores on a college entrance exam is normal, with a mean of 500 and a standard ...

Essentials Of Statistics

Expand each expression in Exercises 122. (4+2x)(42x)

Finite Mathematics

A certain city divides naturally into ten district neighborhoods. How might a real estate appraiser select a sa...

Probability and Statistics for Engineering and the Sciences

Solve each equation for x if 0x2. Give your answers in radians using exact values only. 2cosx+tanx=secx

Trigonometry (MindTap Course List)

In Problems 25-34, use the matrices below. Perform the indicated operations.

Mathematical Applications for the Management, Life, and Social Sciences

How many servings of pie remain if there are 1538 pies in the restaurant and 1212 pies are used?

Elementary Technical Mathematics

In Exercise 29 to 32, use the fact that triangles are similar. As a garage door closes, light is case 6 ft beyo...

Elementary Geometry For College Students, 7e

Convert the following decimals to fractions and reduce to lowest terms. 27.

Contemporary Mathematics for Business & Consumers

Critical Thinking Consider the following events for a driver selected at random from the general population: A ...

Understanding Basic Statistics

Evaluating a Flux Integral In Exercises 25-30, find the flux of F across S, SFNdS where N is the upward unit no...

Calculus: Early Transcendental Functions (MindTap Course List)

A trapezoid has an area of 96 cm2. If the altitude has a length of 8 cm and one base has a length of 9 cm, find...

Elementary Geometry for College Students

TRAFFIC CONTROL The accompanying figure shows the flow of downtown traffic during the rush hours on a typical w...

Finite Mathematics for the Managerial, Life, and Social Sciences

Using Partial Fractions In Exercises 3744, use partial fractions to find the indefinite integral. x2x22x+1dx

Calculus (MindTap Course List)

A surface S is parameterized by x = sin u, y = sin v, z = cos v for (u, v) ∈ D, 0 ≤ u ≤ , 0 ≤ v ≤ . A double in...

Study Guide for Stewart's Multivariable Calculus, 8th

If f(x) = 2e3x, f(x) = a) 2e3x b) 2xe3x c) 6e3x d) 6xe3x

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Cross Product of Unit Vectors In Exercises 3-6, find the cross product of the unit vectors and sketch your resu...

Multivariable Calculus

Consider the following results for independent samples taken from two populations. Sample 1 Sample 2 n1 = 400 n...

Statistics for Business & Economics, Revised (MindTap Course List)

Solve each of the following equations using either the addition, subtraction, or division principle of equality...

Mathematics For Machine Technology

Many companies use a quality control technique called acceptance sampling to monitor incoming shipments of part...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Critical thinking Determine if the statement is true or false. If the statement is false, then correct it and m...

College Algebra (MindTap Course List)

Describe the purpose and responsibilities of the Institutional Review Board (IRB).

Research Methods for the Behavioral Sciences (MindTap Course List)

Suppose that your statistics professor tells you that the scores on a midterm exam were approximately normally ...

Introduction To Statistics And Data Analysis

Explain the role of voluntary participation in informed consent.

Research Methods for the Behavioral Sciences (MindTap Course List)

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