Problem 93 - Calculus Early Transcendentals

Question

A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. The length of the latus rectum of an ellipse centered at the origin is \(2 b^{2} / a=2 b \sqrt{1-e^{2}}\)

Step-by-Step Solution

Answer

The length of the latus rectum of an ellipse centered at the origin is given by the formula: \(2b^2 / a = 2b \sqrt{1 - e^2}\), where 'e' is the eccentricity of the ellipse.

Step 1: Recall the equation of the ellipse centered at the origin

The equation of an ellipse centered at the origin with semi-major axis \(a\) and semi-minor axis \(b\) is given by: \[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\]

Step 2: Identify the foci of the ellipse

The foci of the ellipse are located at points \((\pm ae,0)\), where \(e\) is the eccentricity of the ellipse, which is given by: \[e=\sqrt{1-\frac{b^2}{a^2}}\]

Step 3: Find the equation of a line through one of the foci, perpendicular to the major axis

Let's use the focus located at \((ae,0)\). A line through this point and perpendicular to the major axis will be a vertical line with equation \(x=ae\).

Step 4: Determine two points on the ellipse that lie on the line through the focus

We want to find the intersection points between the vertical line \(x=ae\) and the ellipse. First, let's substitute \(x=ae\) into the equation of the ellipse: \[\frac{(ae)^2}{a^2}+\frac{y^2}{b^2}=1\] Next, simplify and solve for \(y^2\): \[\frac{e^2 a^2}{a^2}+\frac{y^2}{b^2}=1\] \[\frac{y^2}{b^2}=1-e^2\] \[y^2=b^2(1-e^2)\] This gives us two points on the ellipse that lie on the line through the focus: \((ae, \pm b\sqrt{1-e^2})\).

Step 5: Find the distance between the two points

Finally, we will find the distance between the two points \((ae, b\sqrt{1-e^2})\) and \((ae, -b\sqrt{1-e^2})\): The y-coordinates differ by \(2b\sqrt{1-e^2}\) while the x-coordinates are the same. Hence, using distance formula: \[\mathrm{Distance} = 2b\sqrt{1-e^2}\] Thus, the length of the latus rectum of an ellipse centered at the origin is \(2 b^2 / a = 2 b \sqrt{1 - e^2}\).

Key Concepts

Ellipse

An ellipse is a fascinating shape, resembling an elongated circle. Unlike a perfect circle, an ellipse has two axes: the major axis, which is the longest diameter across the figure, and the minor axis, which is the shortest. The minor and major axes intersect at the center of the ellipse.

This shape is defined mathematically by the equation:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]where \(a\) and \(b\) are the semi-major and semi-minor axes, respectively. If the ellipse is centered at the origin, as mentioned in the exercise, these make it easier to plot.

One of the special features of an ellipse is its foci. These are two fixed points situated along the major axis. The sum of the distances from the two foci to any point on the ellipse is constant. This is a unique property of an ellipse and is what defines its shape.

Understanding the significance of the foci in an ellipse is crucial, as they play an essential role in deriving properties such as the latus rectum and eccentricity.

Latus Rectum

The latus rectum of an ellipse is a line segment that is quite unique in its geometry. It is a special focal chord that is perpendicular to the major axis and passes through one of the foci of the ellipse.

This line segment provides insight into the structure of the ellipse. For an ellipse centered at the origin, the length of the latus rectum can be expressed using the formula:\[2b\sqrt{1-e^2}\]This formula arises from considering the vertical line through a focus of the ellipse and calculating the distance between the points where this line intersects the ellipse.

The simplicity of this rule helps in determining other properties of the ellipse and serves as a critical parameter in many mathematical applications and problems related to the ellipse.

Eccentricity

Eccentricity, denoted as \(e\), is a measure of how "stretched" an ellipse is compared to a perfect circle. It is a non-negative number less than 1 for ellipses. The formula to calculate eccentricity is:\[e = \sqrt{1 - \frac{b^2}{a^2}}\]where \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.

An eccentricity of 0 signifies a perfect circle, while values closer to 1 represent increasingly elongated shapes. Eccentricity is critical because it affects the position of the foci and, consequently, other derived attributes such as the latus rectum.

The concept of eccentricity helps us understand the physical nature of orbits in space, as many celestial bodies travel in elliptical orbits with varying eccentricity.

Focal Chord

A focal chord is any line segment that connects two points on the ellipse and passes through one of its foci. These chords are interesting because they depict the geometry involving the ellipse's foci, a feature that distinguishes ellipses from other conic sections.

The focal chord that is perpendicular to the major axis has a specific name: the latus rectum. This special case of a focal chord plays a vital role in understanding the relation between the ellipse's major features like its semimajor axis and eccentricity.

Understanding the focal chord, especially the latus rectum, provides information about the gravitational forces and orbital mechanics that are relevant in astrophysics. With ellipses being foundational in planetary motion, knowing about all chords through a focus enhances conceptual clarity.