Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Vertices \(V(0, \pm 5)\), minor axis of length 3

Step-by-Step Solution

Answer

The equation is \( \frac{x^2}{2.25} + \frac{y^2}{25} = 1 \).

Step 1: Identify the Orientation

Since the vertices of the ellipse are given as \( V(0, \pm 5) \), it indicates that the major axis is vertical. Therefore, the ellipse has the form \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), where \( a \) represents the semi-major axis length.

Step 2: Find the Semi-Major Axis Length

The vertices occur at a distance of 5 units from the center on the y-axis, meaning \( a = 5 \). Thus, the semi-major axis length is 5.

Step 3: Find the Semi-Minor Axis Length

The length of the minor axis is 3, so the semi-minor axis \( b \) is half of this length: \( b = \frac{3}{2} = 1.5 \).

Step 4: Write the Equation of the Ellipse

Substitute \( a = 5 \) and \( b = 1.5 \) into the equation for a vertically oriented ellipse: \( \frac{x^2}{(1.5)^2} + \frac{y^2}{5^2} = 1 \) or simplified, \( \frac{x^2}{2.25} + \frac{y^2}{25} = 1 \).

Key Concepts

Semi-Major Axis

In the context of an ellipse, the term "semi-major axis" refers to half of the longest diameter of the ellipse. When we imagine an ellipse, it's not a perfect circle. It has one longer axis and one shorter axis. The longer one is called the major axis, and half of that length from the center of the ellipse to one end is the semi-major axis.

The semi-major axis is especially important in defining an ellipse because it helps determine the shape's 'stretchiness.' For the ellipse in our exercise, which has its center at the origin, the end points of the major axis on the y-axis are at (0, 5) and (0, -5). This indicates that the full length of the major axis is 10, while the semi-major axis is:

• Half of the full length of the major axis.
• The value of the semi-major axis is 5, as calculated from half of 10.

The length of the semi-major axis is denoted by the letter \( a \) in ellipse equations and is crucial in forming the correct mathematical representation of the ellipse.

Semi-Minor Axis

The semi-minor axis is half the length of the shorter diameter of the ellipse. In an ellipse, while one axis is longer (the major axis), there is also a shorter axis crossing the center—known as the minor axis. The semi-minor axis is half this length, reaching from the center to the edge of the ellipse along the shorter diameter.

In our given exercise, the minor axis has a total length of 3. Therefore, the semi-minor axis is calculated as:

• Half of 3, which equals 1.5.

This shorter measurement emphasizes the narrowing of the ellipse along this axis compared to the major one. The semi-minor axis is expressed using \( b \) in equations for ellipses.

Understanding both the semi-major and semi-minor axes is essential for sketching an accurate ellipse, as they guide how elongated or "oval-shaped" the ellipse appears.

Ellipse Orientation

Orientation of an ellipse defines how the ellipse is placed on the coordinate plane. It determines which axis, the horizontal (x-axis) or the vertical (y-axis), serves as the major or minor axis.

For the ellipse in our problem, the given vertices are at (0, 5) and (0, -5), indicating that the major axis can be found along the y-axis. This orientation tells us:

• The ellipse is vertically oriented because the major axis is vertical.

Therefore, when writing the equation for an ellipse with a vertical orientation, the structure is:\[\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\]where \( a \) is the length of the semi-major axis and \( b \) is the length of the semi-minor axis. Any alteration in orientation, such as a horizontal alignment, would adjust this setup, where \( x \) and \( y \) values swap their respective coefficients in the denominator.

Knowing the orientation is crucial to forming the correct equation and understanding the geometric shape of the ellipse based on the given conditions.