Problem 1 - Algebra 2

Question

Write an equation of an ellipse with the given characteristics. Check your answers. center \((-2,1),\) horizontal major axis of length \(6,\) minor axis of length 4

Step-by-Step Solution

Answer

The equation of the ellipse is \[(x + 2)^2 / 9 + (y - 1)^2 / 4 = 1\]

Step 1: Understand Ellipse Equation

The standard equation of an ellipse is given by \[(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1\] where (h, k) represents the center, and \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. Because the given ellipse has a horizontal major axis, we use \(a\) for the horizontal axis and \(b\) for the vertical axis.

Step 2: Substitution of Values

Substitute \(-2\) for \(h, 1\) for \(k, 3\) for \(a\) (since the length of the major axis is 6, the semi-major length will be 3), and \(2\) for \(b\) (as the length of the minor axis is 4, hence semi-minor length will be 2 ) in the ellipse formula.

Step 3: Write Final Equation

The equation of the ellipse will be \[(x + 2)^2 / 3^2 + (y - 1)^2 / 2^2 = 1\] which simplifies to: \[(x + 2)^2 / 9 + (y - 1)^2 / 4 = 1\]

Key Concepts

Horizontal Major Axis

In an ellipse, the major axis is the longest diameter that passes through the center. When it is horizontal, it stretches across the width of the ellipse. This means that the focal points and the center are aligned horizontally.
To visualize it, imagine a stretched oval lying flat on its side.
The major axis covers the widest part of the ellipse, and defining it as horizontal gives a specific orientation to the ellipse's equation. This aspect affects how we assign values to the semi-major and semi-minor axes in the equation.

• If the major axis is horizontal, the term with 'x' (or the first term) will have the larger denominator.
• In our exercise, the horizontal major axis is essential because it determines where to place the larger value between 'a' and 'b' in the ellipse equation.

Semi-Major Axis Length

The semi-major axis is half the length of the major axis. It is crucial since it helps to determine the proportions of the ellipse. In our case, with a given major axis length of 6, the semi-major axis is 3.
Being the longest radius in an ellipse, the semi-major axis directly impacts the shape and size shown in the equation.

• For our problem, a = 3 because the total major axis is 6.
• This length defines the distance from the center to the furthest edge along the major axis.

Understanding this concept helps give form to the ellipse in a coordinate plane, as it affects the formula's leading denominator.

Semi-Minor Axis Length

The semi-minor axis is half of the minor axis, the shorter diameter across the ellipse. It is perpendicular to the major axis and affects the equation's term related to 'y'.
In the problem, the minor axis is 4, making the semi-minor axis 2.

• The semi-minor axis determines the height of the ellipse.
• When placing it in the equation, b = 2 because the total minor axis is 4.

This value is important in shaping the ellipse's vertical stretch, giving the ellipse its overall oval appearance.

Standard Form of an Ellipse

Ellipses have a standard mathematical representation that simplifies their description. The equation \[(x - h)^2/a^2 + (y - k)^2/b^2 = 1\]is central to understanding their geometry.
The parameters (h, k) represent the center, while a and b are the semi-major and semi-minor axes.

• If the major axis is horizontal, 'a' is associated with 'x', making a greater than b.
• This setup allows you to predict the ellipse’s exact placing and dimensions on a graph.

By substituting specific values, we create an equation that maps the ellipse’s shape accurately.