Problem 27 - College Algebra with Modeling and Visualization

Question

Find an equation of the ellipse, centered at the origin, satisfying the conditions. Bccentricity \(\frac{2}{3},\) horizontal major axis of length 6

Step-by-Step Solution

Answer

The equation is \( \frac{x^2}{9} + \frac{y^2}{5} = 1 \).

Step 1: Identify Conditions

We are given that the eccentricity of the ellipse is \( e = \frac{2}{3} \) and the length of the major axis is 6. The major axis being horizontal indicates \( a > b \). The center of the ellipse is at the origin.

Step 2: Determine Semi-major Axis

The length of the major axis is 6, so the semi-major axis \( a \) is half of that. Thus, \( a = \frac{6}{2} = 3 \).

Step 3: Calculate Semi-minor Axis

For an ellipse, the relationship between semi-major axis \( a \), semi-minor axis \( b \), and eccentricity \( e \) is \( e = \frac{\sqrt{a^2 - b^2}}{a} \). We know \( e = \frac{2}{3} \), so \( \frac{\sqrt{3^2 - b^2}}{3} = \frac{2}{3} \).

Step 4: Solve for Semi-minor Axis

Multiplying both sides by 3 gives \( \sqrt{3^2 - b^2} = 2 \). Squaring both sides gives \( 9 - b^2 = 4 \). Solving for \( b^2 \), we get \( b^2 = 5 \). Therefore, \( b = \sqrt{5} \).

Step 5: Write the Ellipse Equation

The standard form of the ellipse equation with a horizontal major axis is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Substituting \( a = 3 \) and \( b = \sqrt{5} \), the equation becomes \( \frac{x^2}{9} + \frac{y^2}{5} = 1 \).

Key Concepts

Eccentricity

Eccentricity is a measure of how much an ellipse deviates from being a perfect circle. It tells us the shape and degree of elongation of the ellipse. The eccentricity of an ellipse is represented by the symbol \( e \) and can take values between 0 and 1:

• If \( e = 0 \), the ellipse is actually a circle.
• If \( e \) is closer to 1, the ellipse is more elongated.

For the given problem, the eccentricity \( e \) is \( \frac{2}{3} \). This means the ellipse is neither too stretched nor perfectly circular, giving us a nice oval shape. In mathematical terms, for an ellipse with a semi-major axis \( a \) and a semi-minor axis \( b \), the eccentricity is determined by the formula:\[e = \frac{\sqrt{a^2 - b^2}}{a}\] Understanding eccentricity helps us know the ellipse's general appearance without needing a visual representation.

Semi-major axis

The semi-major axis of an ellipse is one of its most important elements. It represents half of the longest diameter of the ellipse and is denoted by \( a \). When you think about the semi-major axis, consider it as the main stretch along the width.

• For a horizontal ellipse, the semi-major axis lies along the x-axis.
• Its length is half of the total major axis.

In this problem, the length of the major axis is 6, therefore, the semi-major axis, \( a \), is calculated as:\[a = \frac{6}{2} = 3\]This means the ellipse stretches 3 units in both positive and negative directions along the x-axis when its center is at the origin. Understanding this concept helps in forming the foundational structure of the ellipse equation.

Semi-minor axis

The semi-minor axis is another critical element of an ellipse. It extends half the distance of the shortest diameter of the ellipse, and is denoted by \( b \). For our ellipse centered at the origin:

• The horizontal ellipse implies the semi-minor axis is oriented vertically.
• The formula \( e = \frac{\sqrt{a^2 - b^2}}{a} \) connects the semi-minor axis to eccentricity and the semi-major axis.

In the given scenario, we already know \( a = 3 \) and \( e = \frac{2}{3} \). Plug these into the formula:\[\frac{\sqrt{3^2 - b^2}}{3} = \frac{2}{3}\]Simplifying, we get:\[\sqrt{9 - b^2} = 2 \9 - b^2 = 4 \b^2 = 5 \b = \sqrt{5}\]This tells us the semi-minor axis is roughly 2.24 units long, adding height to our ellipse.

Equation of ellipse

The equation of an ellipse is the mathematical description of its shape. For an ellipse centered at the origin with a horizontal major axis, the standard form of the equation is:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]This is derived by considering how distances \( x \) and \( y \) change relative to the semi-major axis \( a \) and semi-minor axis \( b \). Substituting our calculated values, \( a = 3 \) and \( b = \sqrt{5} \), we form the equation:\[\frac{x^2}{9} + \frac{y^2}{5} = 1\]This equation represents all points \( (x, y) \) that create the shape of the ellipse. Understanding how \( a \) and \( b \) fit into this formula is crucial for graphing and analyzing ellipses in coordinate geometry.