Problem 26 - Precalculus

Question

Find the standard form of the equation of the ellipse with the given characteristics. Center: (2,-1)\(;\) vertex: \(\left(2, \frac{1}{2}\right) ;\) minor axis of length 2

Step-by-Step Solution

Answer

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

Step 1: Identify coordinates and lengths

For this ellipse, the center is given as (h, k) = (2, -1). The vertex is at \(\left(2, \frac{1}{2}\right)\), which gives us the distance to the center, and thus the semi-major axis length a. Since a = distance from center to vertex, after calculating the distance we get \(a = \frac{1}{2} - (-1) = \frac{3}{2}\). The minor axis length is given as 2, which means the semi-minor axis length (b) is half of that, b = 1.

Step 2: Insert values into the equation

Now we can plug the obtained values into the standard form equation for an ellipse, giving \((x-2)^2/(\frac{3}{2})^2 + (y+1)^2/1^2 = 1\).

Step 3: Simplify the equation

We can then simplify this equation to its complete form, which yields \((x-2)^2/(\frac{9}{4}) + (y+1)^2 = 1\). Multiply the whole equation by 4 to get rid of the fraction, which results in: \(4(x-2)^2/9 + 4(y+1)^2 = 4\) or further simplifying to \((4(x-2)^2/9) + (4(y+1)^2) = 4\).

Key Concepts

Standard Form of an Ellipse

When it comes to understanding ellipses, getting familiar with their equation is crucial. The standard form of an ellipse's equation is based on its orientation and dimensions. For a horizontally oriented ellipse, the standard equation can be written as:

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

In contrast, a vertically oriented ellipse takes on a slightly different form:

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

Here, \((h, k)\) is the center of the ellipse, \(a\) represents the semi-major axis length, and \(b\) denotes the semi-minor axis length.
By substituting the correct values into these equations, the ellipse can be accurately described in a mathematical sense. It's essential to simplify whenever possible to make the equation easy to use for further calculations or graphing.

Ellipse Center

The center of an ellipse is a defining feature and is crucial for its construction. It is the midpoint of the longest and shortest diameters of the ellipse. In mathematical terms, it is denoted as \((h, k)\). This point acts as a reference for plotting the entire ellipse on a coordinate plane.
For the given exercise, the center is established at the coordinates \((2, -1)\). Knowing the center allows you to shift the ellipse properly on the graph and helps in forming the ellipse's standard equation. Remember,

• \(h\) is the x-coordinate of the center.
• \(k\) is the y-coordinate of the center.

Understanding where the center is placed is foundational, as all other measures, like axes and vertices, spring from this point.

Semi-Major and Semi-Minor Axis

An ellipse has two main axes: the semi-major and the semi-minor axes. The semi-major axis represents the longest radius of the ellipse, while the semi-minor axis is the shortest. These help determine the shape and size of the ellipse.
For the ellipse in our exercise:

• The semi-major axis length \(a\) is calculated by the distance from the center to the vertex, which we found to be \(\frac{3}{2}\).
• The semi-minor axis length \(b\) is half of the given minor axis length, so \(b = 1\).

Understanding these axes is critical because they help in delineating the ellipse's boundaries when graphed. Each axis length is plugged into the standard form equation, indicating their proportional influence on the ellipse's geometry.

Vertex of an Ellipse

Vertices are vital for sketching the outline of an ellipse. They are the points where the ellipse touches its axes, marking the endpoints of the semi-major and semi-minor axes. Consequently, each ellipse has four vertices.

• For a horizontally oriented ellipse, its vertices are located at \((h \pm a, k)\) and \((h, k \pm b)\).
• For a vertically oriented ellipse, they appear at \((h, k \pm a)\) and \((h \pm b, k)\).

In this exercise, the vertex \(\left(2, \frac{1}{2}\right)\) provides us the much-needed length of the semi-major axis. By knowing one vertex, you can compute the semi-major axis length, aiding in correctly framing the ellipse's contour. Identifying and using vertices is a significant step in both practical applications and geometric interpretations of ellipses.