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Chapter 10: Problem 51

What happens to an ellipse with major axis \(2 a\) if \(e\) tends to \(1 ?\)

Short Answer

Expert verified
As the eccentricity \(e\) of an ellipse approaches \(1\), the distance between the foci becomes equal to the length of the semi-major axis, resulting in the ellipse transforming into a straight line or a line segment with length equal to the major axis \(2a\).

Step by step solution

01

Definition of ellipse and its parameters

An ellipse is a conic section, which is defined as the set of all points in a plane such that the sum of the distances from two fixed points, called foci (plural of focus), is constant. This constant sum is equal to the length of the major axis, which is given by \(2a\). The parameter \(e\) is the eccentricity of the ellipse, which defines its shape, as well as the position of the foci. The equation of an ellipse is given by: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \(a\) and \(b\) are the lengths of the semi-major axis and the semi-minor axis, respectively.
02

Eccentricity and the distance between the foci

The eccentricity of an ellipse is defined as the ratio of the distance between the foci (denoted as \(2c\)) to the length of the major axis (\(2a\)): \(e = \frac{c}{a} \) Given the relationship \(a^2 - c^2 = b^2\), we can rewrite this as: \(e = \frac{\sqrt{a^2 - b^2}}{a} \) This formula shows us the dependence of the eccentricity on both the semi-major and semi-minor axis lengths.
03

Analyzing the ellipse as \(e\) approaches \(1\)

As \(e\) approaches \(1\), the formula above becomes: \[ \lim_{e \to 1} e = \frac{\sqrt{a^2 - b^2}}{a} \] Let's solve the limit for \(b^2\) keeping the relation \(a^2 - c^2 = b^2\) in mind: \[ \lim_{e \to 1} e = \lim_{e \to 1} \frac{\sqrt{a^2 - b^2}}{a} \] \[ 1 = \frac{\sqrt{a^2 - (a^2 - c^2)}}{a} \] \[ a = \sqrt{c^2} \] \[ a = c \] This means that, when the eccentricity approaches \(1\), the distance between the foci equals the length of the semi-major axis.
04

Conclusion

When the eccentricity of an ellipse approaches \(1\), the distance between the foci becomes equal to the length of the semi-major axis. This results in the ellipse transforming into a straight line, which is also known as a degenerate ellipse or a line segment. This line segment's length is equal to the major axis \(2a\).

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Key Concepts

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

Ellipse Geometry
Ellipses are fascinating shapes in geometry, elegantly represented by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). The values \( a \) and \( b \) are crucial here. They define the semi-major and semi-minor axes of the ellipse. The major axis is the longest diameter across the ellipse, and it stretches through its widest part.

In this geometric structure, two points called "foci" play an essential role. The ellipse's defining property is that the sum of the distances from any point on the ellipse to these foci is constant. These distances can help understand its shape and behavior.

The parameter \( e \), or eccentricity, is a critical element. It affects how stretched the ellipse appears. Smaller eccentricities mean a more circular shape, whereas values approaching 1 flatten the ellipse.
Conic Sections
Conic sections are the curves obtained by intersecting a plane with a double-napped cone. These include circles, ellipses, parabolas, and hyperbolas. An ellipse is formed when the plane cuts through both nappes of the cone, but does not cross the apex.

The beauty of ellipses lies in their uniformity and their peculiar properties, derived from the geometry of conic sections. Each conic can be defined by its eccentricity. For ellipses, the key aspect is that the eccentricity is less than one. This parameter determines how elongated an ellipse is.

Understanding ellipses as conic sections helps reveal broader connections in geometry and astronomy, where ellipses describe planetary orbits, showcasing their natural progression from simple geometric forms.
Degenerate Ellipse Transformation
A degenerate ellipse is a special condition of the elliptical form, appearing as a line segment. This transformation literally "flattens" the ellipse into a single line.

This occurs when the eccentricity \( e \) approaches 1. In this context, the foci become so far apart that they align with the ellipse's major axis length. The equation unfolds in a way that the semi-major and the distance between foci \( 2c \) equal \( 2a \), effectively reducing the ellipse to a "degenerate" line segment.

Degenerate ellipses demonstrate the intriguing possibilities within geometric transformations. They illustrate how continuous change in parameters like eccentricity can lead to a dramatic transition in the shape, enriching our understanding of geometrical behavior.

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Most popular questions from this chapter

Find the surface area of the ellipsoid obtained by revolving the eclipse \(\frac{x^{2}}{a^{2}}+\frac{2}{b^{2}}=1 \quad(0, b

Find the points \((x, y)\) at which the curve has: (a) a horizontal tangent: (b) a vertical tangent. Then sketch the curve. $$x(t)=\sin 2 t, \quad y(t)=\sin t$$

(An interesting property of the sphere) Slice a sphere along two parallel planes a fixed distance apart. Show that the surface area of the band so obtained depends only on the distance between the planes, not on their location.

Find a parametrization $$x \quad x(t), \quad y \quad y(t)\quad t \in[0,1]$$ for the given curve. The line segment from (2,6) to (6,3)

Find the points \((x, y)\) at which the curve has: (a) a horizontal tangent: (b) a vertical tangent. Then sketch the curve. $$x(t)=3 t-t^{3}, \quad y(t)=t+1$$
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