Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

How long is the latus rectum (chord through the focus perpendicular to the major axis) for the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) ?

Step-by-Step Solution

Answer

The latus rectum is \(\frac{2b^2}{a}\) if \(a > b\) or \(\frac{2a^2}{b}\) if \(b > a\).

Step 1: Understanding the Major Axis

For the ellipse given by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we need to first identify which axis is the major axis. If \(a > b\), the major axis is along the x-axis; if \(b > a\), the major axis is along the y-axis.

Step 2: Identify the Focus

For an ellipse centered at the origin, the foci lie along the major axis. The distance of each focus from the center is given by \(c = \sqrt{a^2 - b^2}\) if \(a > b\), or \(c = \sqrt{b^2 - a^2}\) if \(b > a\). These foci are located at \((\pm c, 0)\) or \((0, \pm c)\) depending on the major axis.

Step 3: Determine the Length of the Latus Rectum

The latus rectum for an ellipse is a line segment perpendicular to the major axis that passes through a focus. The length of the latus rectum \(L\) for an ellipse is determined by the formula \(L = \frac{2b^2}{a}\) if \(a > b\), or \(L = \frac{2a^2}{b}\) if \(b > a\).

Step 4: Calculate the Result

Based on the relationship between \(a\) and \(b\), compute the length of the latus rectum using the formula identified in the previous step.

Key Concepts

Latus Rectum

The latus rectum of an ellipse is a straight line segment that is perpendicular to the major axis and passes through one of the foci. Its length is important in understanding the dimensions and properties of an ellipse. To find the length of the latus rectum, we consider the ellipse equation \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]The length of the latus rectum, denoted as \(L\), is calculated differently based on the orientation of the ellipse:

• If \(a > b\), meaning the ellipse is elongated along the x-axis, the latus rectum is given by \(L = \frac{2b^2}{a}\). This ensures that the latus rectum passes perpendicularly through the focus located along the x-axis.
• If \(b > a\), indicating the ellipse stretches more along the y-axis, the latus rectum is calculated as \(L = \frac{2a^2}{b}\).

With these formulas, you can measure how the ellipse is stretched or compressed around its foci.

Major Axis

The major axis is the longest diameter of an ellipse and plays a crucial role in defining its orientation and shape. To determine the major axis, look at the equation of the ellipse: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]The relationship between \(a\) and \(b\) helps identify the major axis:

• If \(a > b\), then the major axis lies along the x-axis, making the ellipse wider horizontally. This corresponds to an orientation where the equation is structured so that the larger base \(a\) contributes to horizontal stretching.
• Conversely, if \(b > a\), the major axis is along the y-axis, resulting in a vertically elongated ellipse.

Understanding which axis is the major one helps in analyzing the shape and positioning of the ellipse.

Focus

The focus (or foci, since there are two) of an ellipse are points located along the major axis that influence how the ellipse curves. The concept of the focus is vital for creating an accurate model of the ellipse.In an equation \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\], the foci are positioned according to the values of \(a\) and \(b\):

• For an ellipse with a major axis along the x-axis (\(a > b\)), each focus is located at \((\pm c, 0)\) where \(c = \sqrt{a^2 - b^2}\).
• If the major axis is aligned with the y-axis (\(b > a\)), the foci are at \((0, \pm c)\) with \(c = \sqrt{b^2 - a^2}\).

Foci are essential for determining the ellipse's geometry and symmetry.

Ellipse Equation

The ellipse equation \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]is a standard mathematical representation, illustrating the relationship between the x and y coordinates in an ellipse.Key components of this equation include:

• \(a\) and \(b\) are the semi-major and semi-minor axes respectively, which dictate the size and orientation of the ellipse.
• The form of the equation shows symmetry centered at the origin \((0,0)\), providing a balanced layout crucial for understanding ellipses.

The equation helps in quickly defining the dimensions and orientation of any standard ellipse, simplifying the process of deriving properties like the major axis, foci, and latus rectum.