Problem 13 - A Graphical Approach to College Algebra

Question

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$\frac{x^{2}}{4}=1-\frac{y^{2}}{9}$$

Step-by-Step Solution

Answer

The equation represents an ellipse.

Step 1: Identify the Standard Form

The given equation is \( \frac{x^2}{4} = 1 - \frac{y^2}{9} \). To identify the conic section, recognize that this equation isn't in the standard form of a conic section yet. Standard forms help classify the conic sections.

Step 2: Rearrange the Equation

Rearrange the given equation to resemble the standard form of known conics. Start by moving \( \frac{y^2}{9} \) to the left side to yield \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \). Now compare this with the standard forms of conic sections to classify it.

Step 3: Compare with Conic Section Forms

Compare the rearranged equation \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \) with the standard form of an ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, both terms are positive, and the equation equals 1, fitting the form of an ellipse.

Step 4: Identify the Conic Section Type

Since the equation fits the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), it represents an ellipse. In this particular form, \( a^2 = 4 \) and \( b^2 = 9 \), confirming the presence of an ellipse.

Key Concepts

Ellipse

An ellipse is a fascinating shape found in geometry and is one of the four classic conic sections. It appears like an elongated circle and is defined by the sum of distances from any of its points to two fixed points, called foci, being constant. Different from a circle, an ellipse has two axes:

• The major axis: The longest diameter that passes through the center and both foci.
• The minor axis: The shortest diameter that is perpendicular to the major axis at the center.

One clear property of ellipses is that they can vary in width and height. This is determined by the values of their semi-major and semi-minor axes, often denoted by the letters 'a' and 'b', respectively. The larger value (whether 'a' or 'b') gives the direction of the ellipse's stretching. Therefore, understanding ellipses involves visualizing this deviation from a perfect circle and recognizing the equation that describes it.

Standard Form of Conic Sections

The standard form of conic sections provides a systematic way to classify different types of curves represented by quadratic equations. Conic sections include circles, ellipses, parabolas, and hyperbolas, each with its specific standard equation. In general, these forms help in predicting the behavior and shape of the graph without necessarily plotting it.

For an ellipse, the standard form of the equation is:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where:

• 'a' represents the semi-major axis.
• 'b' represents the semi-minor axis.

Recognizing that the coefficients of \(x^2\) and \(y^2\) are both positive and the sum equals 1 is crucial to identifying a conic section as an ellipse. If both denominators differ, it indicates whether the ellipse is horizontally or vertically oriented, depending on which quotient (\(a^2\) or \(b^2\)) is larger.

Equation Rearrangement

Rearranging an equation is a valuable skill in mathematics, especially when dealing with conic sections. It's about transforming an equation into a recognizable standard form, which aids in identifying the type of curve it represents. Starting with an unfamiliar form, we manipulate it step by step to fit a known pattern.

In the exercise, the rearrangement begins with moving the term \(\frac{y^2}{9}\) to the left side, yielding:\[ \frac{x^2}{4} + \frac{y^2}{9} = 1 \]This aligns with the standard form of an ellipse. Notice how each term sits comfortably in its respective place, aligning with that standard structure. Thus, rearrangement simplifies comparison to known forms, making it easier to classify conic sections like ellipses, circles, etc., which helps avoid graphing the equation directly.