Problem 25 - Thomas Calculus

Question

Exercises 25 and 26 give information about the foci and vertices of ellipses centered at the origin of the \(x y\) -plane. In each case, find the ellipse's standard-form equation from the given information. $$ \text { Foci: }( \pm \sqrt{2}, 0) \quad \text { Vertices: }( \pm 2,0) $$

Step-by-Step Solution

Answer

The equation of the ellipse is \( \frac{x^2}{4} + \frac{y^2}{2} = 1 \).

Step 1: Identify the Orientation of the Ellipse

Since the foci and vertices are given along the x-axis, the ellipse is oriented horizontally. The standard form of the ellipse's equation for this orientation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

Step 2: Determine the Semi-Major Axis Length

The vertices \( (\pm 2, 0) \) indicate that the semi-major axis length is \( a = 2 \). Therefore, \( a^2 = 4 \).

Step 3: Determine the Distance to the Foci

The foci are given as \( (\pm \sqrt{2}, 0) \), meaning \( c = \sqrt{2} \). The relationship between \( a \), \( b \), and \( c \) for ellipses is \( c^2 = a^2 - b^2 \).

Step 4: Solve for the Semi-Minor Axis Length

Using the equation \( c^2 = a^2 - b^2 \), we substitute the known values: \( (\sqrt{2})^2 = 4 - b^2 \). Simplifying gives \( 2 = 4 - b^2 \), leading to \( b^2 = 2 \).

Step 5: Write the Standard Form Equation

Substitute \( a^2 = 4 \) and \( b^2 = 2 \) into the standard form equation: \( \frac{x^2}{4} + \frac{y^2}{2} = 1 \).

Key Concepts

Ellipses

An ellipse is an interesting geometric shape that resembles an elongated circle. It occurs when a plane cuts through a cone at an angle, and it's one of the conic sections taught in mathematics.
Unlike circles, ellipses have two axes of symmetry: the major and minor axes. These axes determine the lengths and overall shape of the ellipse. The equation that describes an ellipse can be visualized in a standard form, which varies depending on its orientation, either horizontal or vertical.
The standard form of an ellipse's equation is given by:

• Horizontal orientation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
• Vertical orientation: \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\)

Where \(a\) is always the length of the semi-major axis, and \(b\) is the length of the semi-minor axis. The ellipse is centered at the origin if the equation features neither \(x-h\) nor \(y-k\).
This unique shape is widely observed in real-life phenomena, such as planetary orbits, and is integral in fields like astronomy and physics.

Foci and Vertices

The foci and vertices are key characteristics used to define and understand ellipses.
Vertices are points on the ellipse that lie on the axes and farthest from its origin, while foci are distinct points inside the ellipse. Importantly, for an ellipse centered at the origin:

• Vertices on the major axis indicate the maximum distance from the center to the edge of the ellipse.
• The foci indicate focal points around which the ellipse's shape is oriented.

The relationship between the foci, vertices, and the center of an ellipse encapsulates the beauty of mathematical symmetry
The vertices, especially, are crucial in defining the semi-major and semi-minor axes, as they help ascertain the dimensions of the ellipse.
Distances from the center to the foci and vertices are involved in key formulas, like the relationship \( c^2 = a^2 - b^2 \), where:

• \( c \) is the distance to each focus
• \( a \) is the distance to each vertex on the major axis
• \( b \) is the distance to each vertex on the minor axis

This foundational relationship is pivotal in understanding ellipses.

Semi-major and Semi-minor Axes

The semi-major and semi-minor axes are two vital components that govern the shape and size of an ellipse.
The semi-major axis is the longest diameter and goes from one vertex to the opposite vertex through the center of the ellipse. It defines the "stretch" of the ellipse.
Conversely, the semi-minor axis is the shortest diameter. It stretches across the ellipse at a right angle to the semi-major axis, also passing through the center. Together, these axes establish the fundamental proportions of the ellipse.
In the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( a \) (from the semi-major axis) is always larger than \( b \) (from the semi-minor axis), assuming \( a > b \).
This characteristic difference informs the equation's orientation and overall configuration:

• For a horizontally oriented ellipse, the major axis runs along the \( x \)-axis.
• For a vertically oriented ellipse, the major axis aligns with the \( y \)-axis.

Understanding these axes is intuitive once we recognize how they influence the size and symmetry of an ellipse. Together, they help make valuable predictions about the behavior and path of objects following elliptical trajectories.