Problem 31 - Algebra and Trigonometry

Question

In Exercises \(25-36,\) find the standard form of the equation of each ellipse satisfying the given conditions. Major axis horizontal with length \(8 ;\) length of minor axis \(=4\) center; \((0,0)\)

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse is \(\frac{x^2}{16} + \frac{y^2}{4} = 1\).

Step 1: Determine the semiaxes

To calculate the semiaxes, it is required to divide the length of the major and minor axes by 2, as the length of the axes are actually twice the semiaxes. In this case, the length of the major axis is given as 8, so its semi-major axis \(a\) would be \(\frac{8}{2} = 4\). Similarly, the length of the minor axis is given as 4, so its semi-minor axis \(b\) would be \(\frac{4}{2} = 2\).

Step 2: Plug the semiaxes into the standard form

Now that we have both \(a\) and \(b\), we can plug them into the standard form of the ellipse. With \(a=4\) and \(b=2\), our equation becomes \(\frac{x^2}{4^2} + \frac{y^2}{2^2} = 1\) or \(\frac{x^2}{16} + \frac{y^2}{4} = 1\). This is the standard form of the equation for the ellipse with a major axis along the x-axis, minor axis along the y axis, and center at the origin.

Key Concepts

Standard Form

The standard form of an ellipse equation is essential in mathematics for describing the shape and position of an ellipse in the coordinate plane. It's defined based on the orientation of the major axis.
Ellipses can be oriented either horizontally or vertically, and this affects the structure of the standard form equation.
Generally, when a major axis is horizontal, the standard form of the ellipse's equation is:

• \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \)

This form assumes the ellipse is centered at \((h, k)\), where \(a\) represents the semi-major axis' length extending along the x-axis, and \(b\) represents the semi-minor axis' length extending along the y-axis.
In the specific problem, since the center is \((0,0)\), the equation resolves to:

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

With readily identified values, this structure becomes the framework for understanding and deriving other properties of the ellipse.

Semi-Major Axis

The semi-major axis plays a pivotal role in defining the size and orientation of an ellipse. It is half of the longest diameter line through the center, and dictates how far the ellipse stretches along its major direction.
In the given problem, the major axis is specified as 8 units long.
Calculating the semi-major axis involves simply halving this length:

• \( a = \frac{8}{2} = 4 \)

This result confirms that the ellipse extends 4 units along either side of the origin along the x-axis in this example.
Understanding the semi-major axis is critical for sketching and comprehending the ellipse's overall structure and dimensional characteristics.

Semi-Minor Axis

The semi-minor axis is the short distance from the center of the ellipse to its edge along the ellipse's narrower width. Since the ellipse is wider than it is tall in this case, this axis dictates the extent of the ellipse's vertical stretch.
For the exercise at hand, the minor axis is 4 units long. This divides the minor axis to find the semi-minor axis:

• \( b = \frac{4}{2} = 2 \)

Thus, the semi-minor axis reaches 2 units above and below the ellipse's center point \((0,0)\), aligning along the y-axis.
The semi-minor axis tells us how round or elongated the ellipse is when compared to the semi-major axis, essential for visualizing and calculating areas or further characteristics.

Horizontal Major Axis

The horizontal major axis is a defining feature of an ellipse. It indicates that the longest run is from left to right along the x-axis.
In scenarios where the major axis is horizontal, like in this exercise, the coordinates and calculations adjust accordingly.
When an ellipse has a horizontal orientation, your equation reflects this by assigning the larger denominator to \(x^2\). This choice mirrors the extension along the x-axis:

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

Here, \(a\) (being larger) denotes the horizontal major axis' semi-length.
By analyzing the orientation, one can understand the ellipse's spread and its directionality, important for graphing and interpreting. The distinction between horizontal and vertical major axes also assists in identifying the ellipse's appearance and symmetries.