Problem 46 - Algebra 2

Question

Find the equation of each ellipse centered at the origin. $$ \begin{array}{l}{\text { height: } 24 \text { units }} \\ {\text { width: } 36 \text { units }}\end{array} $$

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^2}{324} + \frac{y^2}{144} = 1\)

Step 1: Identify ellipse parameters

Given the height and width of the ellipse, calculate half of each to obtain the semi-major axis (half the width) and semi-minor axis (half the height). In this instance, the semi-major axis (a) is 18 units (i.e., half of 36 units), and the semi-minor axis (b) is 12 units (i.e., half of 24 units).

Step 2: Plug ellipse parameters into the standard equation

Now that you have determined the semi-major and semi-minor axises, insert the values into the equation of the ellipse at the origin. The equation will appear as follows: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, a = 18 units and b = 12 units.

Step 3: Simplify the equation

As a result, the simplified equation will appear as follows: \(\frac{x^2}{324} + \frac{y^2}{144} = 1\)

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double napped cone. This interaction can result in different types of shapes such as circles, ellipses, parabolas, and hyperbolas. Each shape has its own unique set of properties and equations.
Ellipses, like the one in our exercise, are formed when the plane cuts through the cone at an angle that is not perpendicular to its base. Understanding the properties and equations of ellipses is important because they frequently appear in real-world applications such as planetary orbits and optics.
The key parameters that describe an ellipse include the lengths of its semi-major and semi-minor axes. These determine the ellipse's overall shape and orientation.

Semi-Major Axis

The semi-major axis is the longest radius of an ellipse, stretching from its center to the farthest edge. In our exercise, it is half of the ellipse's width, since the ellipse is aligned with the coordinate axes and has its center at the origin. Given the width of 36 units, the semi-major axis measures 18 units.
The semi-major axis is a crucial parameter for the ellipse's equation. It defines the stretch of the ellipse along the x-axis if it is longer than the semi-minor axis. This feature directly influences the formula \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) represents the semi-major axis length.
Understanding this concept helps to easily differentiate ellipses with varying orientations and sizes.

Semi-Minor Axis

Opposite to the semi-major axis, the semi-minor axis is the shortest radius of an ellipse. It extends from the center of the ellipse to the nearest edge. In our given example, the height of 24 units leads to a semi-minor axis of 12 units (half of the height), running along the y-axis due to the way the ellipse is oriented.
Much like the semi-major axis, the semi-minor axis is integral to calculating the standard form of the ellipse equation. This vector is used to calculate \(b^2\) in the expression \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(b\) is the semi-minor axis length. The ratio between the semi-major and semi-minor axes indicates how elongated the ellipse appears.
It’s important to recognize how these axes affect the general shape and dimensionality of the ellipse.

Cartesian Coordinates

Cartesian coordinates offer a method to describe the position of points in a plane using two numerical values: usually represented as \(x\) and \(y\). This system is fundamental when working with ellipses centered at the origin, as our problem demonstrates.
The ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) utilizes \((x, y)\) points on the Cartesian plane to map out the curve of the ellipse. The center of the ellipse in this system is situated at \( (0,0)\), which provides a symmetric look, simplifying calculations and understanding of the shape.
Mastering the use of Cartesian coordinates is essential when dealing with conic sections, as it allows precise mathematical analyses and plotting of these fascinating geometric figures.