Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Find the eccentricity e of each ellipse or hyperbola. $$12 x^{2}+9 y^{2}=36$$

Step-by-Step Solution

Answer

The eccentricity is \(\frac{1}{2}\).

Step 1: Rewrite the Equation

First, we need to rewrite the given equation in standard form. The equation given is \(12x^2 + 9y^2 = 36\). Divide all terms by 36 to simplify it:\[\frac{12x^2}{36} + \frac{9y^2}{36} = \frac{36}{36}\]This simplifies to:\[\frac{x^2}{3} + \frac{y^2}{4} = 1\]

Step 2: Identify the Ellipse Parameters

Identify the coefficients and the denominators from the equation \(\frac{x^2}{3} + \frac{y^2}{4} = 1\). Here, \(a^2 = 4\) and \(b^2 = 3\), so \(a = 2\) and \(b = \sqrt{3}\). Since \(a > b\), the orientation is horizontal.

Step 3: Calculate the Eccentricity

To find the eccentricity \(e\), use the formula for an ellipse: \[ e = \sqrt{1 - \frac{b^2}{a^2}} \]Substituting the values, \(e = \sqrt{1 - \frac{3}{4}} = \sqrt{\frac{1}{4}} = \frac{1}{2}\).

Step 4: Verify the Ellipse Type and Parameters

Ensure that the given equation represents an ellipse since \(e < 1\). The equation \(12x^2 + 9y^2 = 36\) reduces to an ellipse form confirming our calculations. The values \(a = 2\) and \(b = \sqrt{3}\) remain consistent with these deductions.

Key Concepts

Ellipse

An ellipse is a fascinating shape that you often encounter in math. It looks like a stretched circle and has two main axes: the major and minor axes.
This curve appears in many real-world applications, from planets' orbits to architectural designs.
Here are a few points to know about ellipses:

• Ellipses have two fixed points called "foci." If you add up the distances from any point on the ellipse to these foci, you'll get a constant sum.
• The longer axis of an ellipse is called the major axis, and the shorter one is the minor axis.
• Ellipses have symmetrical shapes, meaning they look the same on both sides of their center.

Ellipses are described mathematically by an equation like \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively.
Learning about ellipses helps us understand more about navigation, music, and even nature.

Eccentricity

Eccentricity measures how much an ellipse deviates from being a circle. It is denoted by the letter \( e \).
An ellipse with zero eccentricity is a perfect circle.
For ellipses, the eccentricity is calculated using the formula:\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]

• Here, \( a \) is the semi-major axis, and \( b \) is the semi-minor axis.
• The value of \( e \) always lies between 0 and 1 for ellipses.
• The closer \( e \) is to zero, the more circular the ellipse is.
• The closer \( e \) is to one, the more elongated the ellipse becomes.

In the problem, when we calculated the eccentricity, we found it to be \( \frac{1}{2} \, \) detailing that the ellipse is moderately oval-shaped.
Understanding eccentricity gives insight into how shapes in our universe, like orbits, differ from perfect circles.

Standard Form of Ellipse

The standard form of an ellipse equation is fundamental for identifying its characteristics, such as orientation and axes.
The equation typically appears as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
Here is why this form is so important:

• The values \( a \) and \( b \) are directly related to the ellipse’s dimensions.
• Whether \( a^2 \) or \( b^2 \) is larger determines the major axis.
• If \( a > b \, \) the ellipse is horizontally oriented, whereas \( b > a \) makes it vertical.

Converting an ellipse equation to this standard form helps easily find important parameters such as the center, axes, and eccentricity.
In our specific example, converting the given equation into standard form helped us deduce that the ellipse is oriented horizontally, with semi-major and semi-minor axes calculated readily.
Recognizing the standard form of an ellipse ensures accurate understanding and representation of its geometric properties.