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Chapter 3: Problem 7

What is an ellipse? Define its foci, semimajor axis, and eccentricity.

Short Answer

Expert verified
An ellipse is an elongated circle with foci, a semimajor axis, and eccentricity that describe its shape.

Step by step solution

01

Defining an Ellipse

An ellipse is a geometric shape that looks like an elongated circle. It can be defined as the set of all points such that the sum of the distances from two fixed points, called foci, is constant.
02

Understanding the Foci

The foci (plural for focus) of an ellipse are two specific points located along its longest axis. The unique property of an ellipse is that, for any point on its surface, the total distance to the two foci remains the same.
03

Identifying the Semimajor Axis

The semimajor axis of an ellipse is the longest radius or half of the longest diameter. It stretches from the center of the ellipse to one end of the ellipse along its longest side.
04

Calculating Eccentricity

Eccentricity is a number that describes the shape of the ellipse. It is calculated using the formula: \( e = \frac{c}{a} \), where \( c \) is the distance from the center to one focus, and \( a \) is the length of the semimajor axis. An eccentricity of 0 means a circle, while values closer to 1 mean a more elongated shape.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Foci of Ellipse
An ellipse is known for its unique feature where two special points, termed as foci (pronounced 'foh-sigh'), play a vital role in its formation. Imagine placing two pins on a sheet of paper, which act as these foci, and then using a flexible string to draw the ellipse. The most interesting property about foci is that the collective distance from any point on the ellipse's outline to both foci is constant.
  • This means that no matter where you are on the edge of the ellipse, if you add up the distances to each focus, it will never change.
  • This constant sum creates the stretched, oval shape characteristic of ellipses, quite different from a perfect circle.
The foci reside on what is known as the major axis, which we will discuss in the next section.
Semimajor Axis
The semimajor axis is a fundamental component in understanding the size and stretch of an ellipse. It is half of the longest diameter of the ellipse and extends from its center to one endpoint along the longest side.
  • Think of it as the longest "radius" that starts at the center of the ellipse and ends at the edge. Its full length is often referred to as the major axis.
  • The importance of the semimajor axis lies in its use in mathematical formulas and calculations about the ellipse's proportions.
If the semimajor axis is relatively short, the ellipse will appear more like a circle, whereas it will look more elongated if the semimajor axis is long. This degree of elongation leads us to consider a critical property called eccentricity.
Eccentricity Definition
Eccentricity is a numerical measure that describes how "stretched" or elongated an ellipse is. It's calculated using the simple formula: \( e = \frac{c}{a} \), where \( c \) is the distance from the center to one of the foci, and \( a \) is the length of the semimajor axis.
  • An ellipse with an eccentricity of 0 is quite special; it is a perfect circle with no elongation whatsoever.
  • As the value of eccentricity approaches 1, the ellipse stretches more, resembling a more flattened shape.
This concept of eccentricity is crucial for differentiating between the various conic sections, like circles and ellipses, and helps in understanding the spatial properties of these shapes.

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Most popular questions from this chapter

Choose the best answer to each of the following. Explain your reasoning with one or more complete sentences. According to Kepler's third law, (a) Mercury travels fastest in the part of its orbit in which it is closest to the Sun. (b) Jupiter orbits the Sun at a faster speed than Saturn. (c) all the planets have nearly circular orbits.

In what way is scientific thinking natural to all of us, and how does modern science build upon this everyday type of thinking?

Choose the best answer to each of the following. Explain your reasoning with one or more complete sentences. Earth is closer to the Sun in January than in July. Therefore, in accord with Kepler's second law, (a) Earth travels faster in its orbit around the Sun in July than in January. (b) Earth travels faster in its orbit around the Sun in January than in July. (c) it is summer in January and winter in July.

Be sure to show all calculations clearly and state your final answers in complete sentences. The dwarf planet Eris orbits the Sun every 557 years. What is its average distance (semimajor axis) from the Sun? How does its average distance compare to that of Pluto?

Be sure to show all calculations clearly and state your final answers in complete sentences. You are an astronomer on planet Nearth, which orbits a distant star. It has recently been accepted that Nearth is spherical in shape, though no one knows its size. One day, while studying in the library of Alectown, you learn that on the equinox your sun is directly overhead in the city of Nyene, located \(1000 \mathrm{km}\) due north of you. On the equinox, you go outside in Alectown and observe that the altitude of your sun is \(80^{\circ} .\) What is the circumference of Nearth? (Hint: Apply the technique used by Eratosthenes to measure Earth's circumference.)
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