Chapter 12.6, Problem 43E

### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

Chapter
Section

### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

# Orbit of the EarthThe polar equation of an ellipse can be expressed in terms of its eccentricity e and the length a of its major axis(a) Show that the polar equation of an ellipse with directrix x = − d can be written in the form r = a ( 1 − e 2 ) 1 − e cos θ [Hint: Use the relation a 2 = e 2 d 2 / ( 1 − e 2 ) 2 given in the proof on page 869.](b) Find an approximate polar equation for the elliptical orbit of the earth around the sun (at one focus) given that the eccentricity is about 0.017 and length of the major axis is about 2.99 × 10 8 km .

To determine

a)

To prove:

The polar equation of an ellipse with directrix x=d can be written in the form r=a(1e2)1ecosθ.

Explanation

Given:

The polar equation of an ellipse with directrix x=d.

Approach:

Use the relation a2=e2d2/(1e2)2… (1)

Where a is length of major axis.

Calculation:

Since,

The polar equation of an ellipse with directrix x=d is

r=ed1ecosθ… (2)

From equation (1),

ed=a(1

To determine

b)

To find:

An approximate polar equation for elliptical orbit of earth around the sun (at one focus).

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